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Tj 1 0 0 1 238.32 633.60 Tm <7265717569726520> Tj 1 0 0 1 275.04 633.36 Tm <7468617420> Tj 1 0 0 1 295.92 633.36 Tm <6561636820> Tj 1 0 0 1 321.36 633.36 Tm <706F737369626C6520> Tj 1 0 0 1 363.60 633.12 Tm <6576656E7420> Tj 1 0 0 1 392.88 633.12 Tm <6861766520> Tj 1 0 0 1 419.04 632.88 Tm <616E20> Tj 1 0 0 1 433.44 632.88 Tm <657175616C20> Tj 1 0 0 1 462.48 632.64 Tm <6368616E636520> Tj 1 0 0 1 499.20 632.64 Tm <6F6620> Tj 1 0 0 1 512.16 632.64 Tm <6F6363756D6E732E> Tj 0.96 Tc 1 0 0 1 63.84 598.32 Tm <50282C7B2920> Tj 1 0 0 1 89.76 600.00 Tm <3D> Tj 0.96 Tc 1 0 0 1 118.80 605.76 Tm <6E756D62657220> Tj 1 0 0 1 158.40 605.76 Tm <6F6620> Tj 1 0 0 1 173.04 606.00 Tm <7761797320> Tj 1 0 0 1 202.32 605.52 Tm <6576656E7420> Tj 1 0 0 1 231.12 605.52 Tm <4120> Tj 1 0 0 1 242.16 605.52 Tm <6D616C276F63637572> Tj 0.72 Tc 1 0 0 1 101.52 588.48 Tm <746F74616920> Tj 1 0 0 1 124.56 588.48 Tm <6E756D62657220> Tj 1 0 0 1 164.40 588.24 Tm <6F6620> Tj 1 0 0 1 178.32 588.24 Tm <646966666272656E7420> Tj 1 0 0 1 222.48 588.00 Tm <6F7574636F6D657320> Tj 1 0 0 1 270.00 587.76 Tm <6F6F737369626C65> Tj 0.96 Tc 1 0 0 1 61.68 558.00 Tm <466F7220> Tj 1 0 0 1 80.88 558.00 Tm <6578616D706C652E20> Tj 1 0 0 1 127.68 557.76 Tm <6C6574277320> Tj 1 0 0 1 152.16 557.76 Tm <73617920> Tj 1 0 0 1 171.12 557.76 Tm <4920> Tj 1 0 0 1 178.32 557.76 Tm <77616E7420> Tj 1 0 0 1 204.72 557.76 Tm <746F20> Tj 1 0 0 1 217.44 557.52 Tm <6B6E6F7720> Tj 1 0 0 1 246.48 557.52 Tm <74686520> Tj 1 0 0 1 264.00 557.76 Tm <70726F626162696C69747920> Tj 1 0 0 1 319.92 557.28 Tm <6F6620> Tj 1 0 0 1 332.88 557.28 Tm <726F6C6C696E6720> Tj 1 0 0 1 368.16 557.28 Tm <6120> Tj 1 0 0 1 376.56 558.24 Tm <22342227757669746820> Tj 1 0 0 1 420.96 557.04 Tm <6120> Tj 1 0 0 1 429.36 557.04 Tm <6661697220> Tj 1 0 0 1 449.04 556.80 Tm <6469652E> Tj 0.96 Tc 1 0 0 1 384.96 604.56 Tm <54686520> Tj 1 0 0 1 407.28 604.56 Tm <6E6F746174696F6E20> Tj 1 0 0 1 450.00 606.00 Tm <22502841292220> Tj 1 0 0 1 486.96 604.32 Tm <6D65616E73> Tj 0.96 Tc 1 0 0 1 385.44 591.84 Tm <2274686520> Tj 1 0 0 1 408.48 590.40 Tm <70726F626162696C69747920> Tj 1 0 0 1 464.64 589.92 Tm <6F6620> Tj 1 0 0 1 477.60 589.92 Tm <6576656E7420> Tj 1 0 0 1 506.88 589.92 Tm <412E22> Tj 0.96 Tc /F3 10.0 Tf 1 0 0 1 114.48 526.80 Tm <6E2C20> Tj 1 0 0 1 134.88 526.80 Tm <72722E20> Tj 1 0 0 1 167.52 527.28 Tm <2C2C20> Tj 1 0 0 1 173.76 526.80 Tm <692C5C20> Tj 1 0 0 1 201.60 529.20 Tm <6E756D62657220> Tj 1 0 0 1 240.72 528.96 Tm <6F6620> Tj 1 0 0 1 254.88 529.20 Tm <7761797320> Tj 1 0 0 1 284.40 528.96 Tm <7468617420> Tj 1 0 0 1 305.28 528.96 Tm <6120> Tj 1 0 0 1 313.20 530.16 Tm <22342220> Tj 1 0 0 1 330.00 528.96 Tm <6D617620> Tj 1 0 0 1 352.56 528.72 Tm <6F63637572> Tj 1.44 Tc 1 0 0 1 114.48 521.76 Tm <74276C20> Tj 1 0 0 1 124.80 521.76 Tm <726F6C6C6C6E737520> Tj 1 0 0 1 171.60 521.76 Tm <2B20> Tj 1 0 0 1 184.80 522.00 Tm <6C3D> Tj 0.72 Tc 1 0 0 1 219.60 511.92 Tm <6E756D62657220> Tj 1 0 0 1 258.00 511.68 Tm <6F6620> Tj 1 0 0 1 271.92 511.68 Tm <646966662D6572656E7420> Tj 1 0 0 1 315.84 511.44 Tm <6F7572636F6D6573> Tj 0.96 Tc /F3 11.0 Tf 1 0 0 1 439.44 530.88 Tm <4E6F746520> Tj 1 0 0 1 466.08 530.64 Tm <7468617420> Tj 1 0 0 1 487.20 530.64 Tm <6120> Tj 1 0 0 1 495.60 536.40 Tm <2727342220> Tj 1 0 0 1 515.28 530.64 Tm <63616E20> Tj 1 0 0 1 535.20 530.64 Tm <6F6E6C79> Tj 0.96 Tc 1 0 0 1 439.44 516.96 Tm <6F6363757220> Tj 1 0 0 1 469.20 516.96 Tm <696E20> Tj 1 0 0 1 481.44 516.72 Tm <6F6E6520> Tj 1 0 0 1 502.32 516.96 Tm <7761762E20> Tj 1 0 0 1 528.72 516.72 Tm <73696E636520> Tj 1 0 0 1 555.84 516.48 Tm <7468657265> Tj 0.96 Tc 1 0 0 1 439.44 502.80 Tm <697320> Tj 1 0 0 1 450.48 502.80 Tm <6F6E697920> Tj 1 0 0 1 474.72 502.56 Tm <6F6E6520> Tj 1 0 0 1 495.36 504.00 Tm <22342220> Tj 1 0 0 1 515.28 502.56 Tm <6F6E20> Tj 1 0 0 1 530.64 502.56 Tm <6120> Tj 1 0 0 1 538.80 502.56 Tm <6469652E> Tj 0.96 Tc 1 0 0 1 439.20 488.88 Tm <4E6F746520> Tj 1 0 0 1 466.08 488.64 Tm <616C736F20> Tj 1 0 0 1 488.16 488.64 Tm <7468617420> Tj 1 0 0 1 509.52 488.64 Tm <74686520> Tj 1 0 0 1 527.52 488.64 Tm <6E756D62657220> Tj 1 0 0 1 566.88 488.40 Tm <6F66> Tj 0.72 Tc 1 0 0 1 439.44 474.72 Tm <646966742D6572656E7420> Tj 1 0 0 1 483.60 474.72 Tm <6F7574636F6D657320> Tj 1 0 0 1 533.28 474.48 Tm <697320> Tj 1 0 0 1 544.32 474.48 Tm <362E20> Tj 1 0 0 1 556.32 474.24 Tm <73726E6365> Tj 0.96 Tc 1 0 0 1 439.44 461.04 Tm <6120> Tj 1 0 0 1 447.60 460.80 Tm <64696520> Tj 1 0 0 1 465.36 460.80 Tm <68617320> Tj 1 0 0 1 484.56 460.80 Tm <73697820> Tj 1 0 0 1 502.08 460.80 Tm <73696465732E> Tj 0.72 Tc 1 0 0 1 60.96 429.12 Tm <457870656E6D656E74616C697920> Tj 1 0 0 1 138.00 428.88 Tm <64637465726D696E656420> Tj 1 0 0 1 195.60 429.12 Tm <70726F626162696C697469657320> Tj 1 0 0 1 258.72 428.64 Tm <617265> Tj 0.96 Tc 1 0 0 1 60.96 414.96 Tm <6F6273656E2C652920> Tj 1 0 0 1 105.36 414.96 Tm <7365766572616C20> Tj 1 0 0 1 142.32 414.96 Tm <747269616C7320> Tj 1 0 0 1 169.92 414.96 Tm <616E6420> Tj 1 0 0 1 190.32 414.96 Tm <636F756E7420> Tj 1 0 0 1 220.32 414.96 Tm <686F727620> Tj 1 0 0 1 243.60 414.72 Tm <6D616E79> Tj 0.72 Tc 1 0 0 1 276.24 428.40 Tm <626173656420> Tj 1 0 0 1 306.48 428.40 Tm <6F6E20> Tj 1 0 0 1 321.84 428.40 Tm <6D616E7920> Tj 1 0 0 1 351.12 428.40 Tm <6F62736572766174696F6E7320> Tj 1 0 0 1 415.44 428.16 Tm <6F7220> Tj 1 0 0 1 428.64 428.16 Tm <746E616C732E20> Tj 1 0 0 1 462.00 428.16 Tm <576520> Tj 1 0 0 1 481.68 428.16 Tm <776F756C6420> Tj 1 0 0 1 515.28 428.16 Tm <706572666F726D20> Tj 1 0 0 1 556.80 429.60 Tm <286F72> Tj 0.72 Tc 1 0 0 1 273.12 414.72 Tm <74696D657320> Tj 1 0 0 1 302.40 414.72 Tm <776520> Tj 1 0 0 1 319.44 414.72 Tm <67657420> Tj 1 0 0 1 337.20 414.48 Tm <6120> Tj 1 0 0 1 345.12 414.48 Tm <6465736972656420> Tj 1 0 0 1 382.80 414.48 Tm <726573756C742E> Tj 0.96 Tc 1 0 0 1 62.64 378.48 Tm <5028412920> Tj 1 0 0 1 88.80 380.16 Tm <3D> Tj 0.96 Tc 1 0 0 1 100.08 386.16 Tm <6E756D62657220> Tj 1 0 0 1 138.72 386.16 Tm <6F6620> Tj 1 0 0 1 153.36 386.16 Tm <74696D657320> Tj 1 0 0 1 181.68 386.16 Tm <6576656E7420> Tj 1 0 0 1 210.48 386.16 Tm <4120> Tj 1 0 0 1 221.52 385.92 Tm <6F6363757273> Tj 0.96 Tc /F3 10.0 Tf 1 0 0 1 125.04 369.12 Tm <746F74616C20> Tj 1 0 0 1 149.52 369.12 Tm <6E756D62657220> Tj 1 0 0 1 188.88 368.88 Tm <6F6620> Tj 1 0 0 1 203.04 368.88 Tm <746E616C73> Tj 0.72 Tc /F3 11.0 Tf 1 0 0 1 60.48 341.76 Tm <466F7220> Tj 1 0 0 1 79.92 341.52 Tm <6578616D706C652E20> Tj 1 0 0 1 126.72 341.52 Tm <4920> Tj 1 0 0 1 133.92 341.52 Tm <6D617920> Tj 1 0 0 1 158.16 341.52 Tm <66696E6420> Tj 1 0 0 1 180.72 341.52 Tm <74686520> Tj 1 0 0 1 198.48 341.76 Tm <70726F626162696C6974792220> Tj 1 0 0 1 253.92 341.28 Tm <6F6620> Tj 1 0 0 1 266.88 341.28 Tm <6D616B696E6720> Tj 1 0 0 1 306.00 341.04 Tm <6120> Tj 1 0 0 1 314.40 341.04 Tm <6672656520> Tj 1 0 0 1 336.00 341.04 Tm <7468726F7520> Tj 1 0 0 1 367.20 341.04 Tm <696E> Tj 0.72 Tc 1 0 0 1 60.72 327.84 Tm <6261736B657462616C6C20> Tj 1 0 0 1 112.32 327.84 Tm <627920> Tj 1 0 0 1 126.96 328.08 Tm <676F696E6720> Tj 1 0 0 1 157.92 327.60 Tm <6F757420> Tj 1 0 0 1 176.64 327.60 Tm <616E6420> Tj 1 0 0 1 197.28 327.60 Tm <617474656D7074696E6720> Tj 1 0 0 1 251.28 327.60 Tm <6120> Tj 1 0 0 1 259.20 327.60 Tm <6672656520> Tj 1 0 0 1 281.04 327.36 Tm <7468726F7720> Tj 1 0 0 1 312.24 327.36 Tm <6D616E7620> Tj 1 0 0 1 341.76 327.12 Tm <74696D65732E> Tj 0.96 Tc 1 0 0 1 60.24 314.16 Tm <5468656E20> Tj 1 0 0 1 88.08 314.16 Tm <74686520> Tj 1 0 0 1 105.84 314.16 Tm <70726F626162696C69747920> Tj 1 0 0 1 162.00 313.92 Tm <72733A> Tj 0.72 Tc 1 0 0 1 186.72 284.88 Tm <6E756D62657220> Tj 1 0 0 1 226.08 284.88 Tm <6F6620> Tj 1 0 0 1 240.00 284.88 Tm <74696D657320> Tj 1 0 0 1 268.08 284.88 Tm <4920> Tj 1 0 0 1 274.56 284.64 Tm <6D616B6520> Tj 1 0 0 1 302.40 284.64 Tm <6120> Tj 1 0 0 1 310.32 284.64 Tm <6672656520> Tj 1 0 0 1 331.92 284.64 Tm <7468726F77> Tj 0.96 Tc /F3 10.0 Tf 1 0 0 1 62.88 277.68 Tm <50746D616B696E6520> Tj 1 0 0 1 111.36 277.68 Tm <6120> Tj 1 0 0 1 119.52 277.68 Tm <6672636320> Tj 1 0 0 1 140.64 277.68 Tm <7468726F722C7220> Tj 1 0 0 1 169.92 276.48 Tm <3B20> Tj 1 0 0 1 175.92 279.36 Tm <3D> Tj 0.72 Tc /F3 11.0 Tf 1 0 0 1 215.28 267.84 Tm <746F74616C20> Tj 1 0 0 1 238.56 267.84 Tm <6E756D62637220> Tj 1 0 0 1 277.44 267.60 Tm <6F6620> Tj 1 0 0 1 291.36 267.60 Tm <617474656D6F7473> Tj 0.72 Tc 1 0 0 1 60.00 238.56 Tm <5468697320> Tj 1 0 0 1 84.48 238.56 Tm <697320> Tj 1 0 0 1 95.52 238.56 Tm <7265616C6C7620> Tj 1 0 0 1 125.76 238.32 Tm <616E20> Tj 1 0 0 1 140.64 238.56 Tm <617070726F78746D6174696F6E20> Tj 1 0 0 1 214.08 238.08 Tm <6265636175736520> Tj 1 0 0 1 254.64 238.32 Tm <696620> Tj 1 0 0 1 265.20 238.08 Tm <4920> Tj 1 0 0 1 272.16 238.56 Tm <676F20> Tj 1 0 0 1 287.04 238.08 Tm <6F757420> Tj 1 0 0 1 305.52 238.08 Tm <74686520> Tj 1 0 0 1 323.28 238.08 Tm <6E65787420> Tj 1 0 0 1 347.04 238.08 Tm <6461312720> Tj 1 0 0 1 367.44 238.08 Tm <616E6420> Tj 1 0 0 1 387.84 237.84 Tm <646F20> Tj 1 0 0 1 402.96 237.84 Tm <74686520> Tj 1 0 0 1 420.96 237.84 Tm <73616D6520> Tj 1 0 0 1 448.56 237.84 Tm <657870656E6D656E742C20> Tj 1 0 0 1 509.04 237.60 Tm <4920> Tj 1 0 0 1 516.48 237.60 Tm <6D617920> Tj 1 0 0 1 540.00 237.84 Tm <67657420> Tj 1 0 0 1 557.76 237.36 Tm <61> Tj 0.72 Tc 1 0 0 1 60.24 224.64 Tm <646966666572656E7420> Tj 1 0 0 1 104.40 224.88 Tm <70726F62616269696974792E20> Tj 1 0 0 1 166.56 224.64 Tm <497420> Tj 1 0 0 1 177.12 224.40 Tm <6D616B657320> Tj 1 0 0 1 211.20 224.16 Tm <73656E736520> Tj 1 0 0 1 239.76 224.40 Tm <7468617420> Tj 1 0 0 1 260.64 224.16 Tm <74686520> Tj 1 0 0 1 278.16 224.16 Tm <6D6F726520> Tj 1 0 0 1 306.00 224.16 Tm <617474656D70747320> Tj 1 0 0 1 349.92 224.16 Tm <4920> Tj 1 0 0 1 356.64 223.92 Tm <7472792C20> Tj 1 0 0 1 376.08 223.92 Tm <74686520> Tj 1 0 0 1 393.84 223.92 Tm <636C6F73657220> Tj 1 0 0 1 426.00 223.92 Tm <74686520> Tj 1 0 0 1 443.76 223.92 Tm <657870656E6D656E74616C20> Tj 1 0 0 1 509.76 223.92 Tm <70726F626162696C697479> Tj 0.72 Tc 1 0 0 1 60.24 210.96 Tm <7520> Tj 1 0 0 1 68.88 210.72 Tm <696C6C20> Tj 1 0 0 1 81.84 210.96 Tm <67657420> Tj 1 0 0 1 99.36 210.72 Tm <746F20> Tj 1 0 0 1 111.60 210.72 Tm <74686520> Tj 1 0 0 1 129.36 210.48 Tm <61637475616C20> Tj 1 0 0 1 161.76 210.96 Tm <70726F626162696C6974792E20> Tj 1 0 0 1 223.44 210.24 Tm <546F20> Tj 1 0 0 1 240.00 210.24 Tm <756E6465727374616E6420> Tj 1 0 0 1 294.96 210.24 Tm <7468697320> Tj 1 0 0 1 315.84 210.24 Tm <6964656120> Tj 1 0 0 1 338.88 210.00 Tm <7468617420> Tj 1 0 0 1 359.76 210.00 Tm <6D6F726520> Tj 1 0 0 1 387.60 210.00 Tm <747269616C7320> Tj 1 0 0 1 414.96 209.76 Tm <6C6561647320> Tj 1 0 0 1 442.80 209.76 Tm <746F20> Tj 1 0 0 1 455.28 209.76 Tm <6120> Tj 1 0 0 1 463.44 209.52 Tm <62657474657220> Tj 1 0 0 1 494.16 209.76 Tm <617070726F72696D6172696F6E2E> Tj 0.72 Tc 1 0 0 1 60.00 196.80 Tm <696D6167696E6520> Tj 1 0 0 1 101.76 196.80 Tm <74686520> Tj 1 0 0 1 119.04 196.80 Tm <6378706572696D656E7420> Tj 1 0 0 1 177.12 196.56 Tm <6F6620> Tj 1 0 0 1 190.08 196.56 Tm <666C697070696E6720> Tj 1 0 0 1 231.36 196.56 Tm <6120> Tj 1 0 0 1 239.52 196.56 Tm <636F6C6E2E> Tj 0.72 Tc 1 0 0 1 59.76 169.20 Tm <4265636175736520> Tj 1 0 0 1 102.96 168.96 Tm <746865726520> Tj 1 0 0 1 129.84 168.96 Tm <61726520> Tj 1 0 0 1 147.84 168.96 Tm <74776F20> Tj 1 0 0 1 169.44 168.96 Tm <736964657320> Tj 1 0 0 1 196.56 169.68 Tm <2868656164732F7461696C732920> Tj 1 0 0 1 257.76 168.72 Tm <6F6E20> Tj 1 0 0 1 272.88 168.72 Tm <6120> Tj 1 0 0 1 281.04 168.72 Tm <6661697220> Tj 1 0 0 1 300.72 168.48 Tm <636F696E2C20> Tj 1 0 0 1 327.60 168.48 Tm <74686520> Tj 1 0 0 1 345.36 168.48 Tm <7468656F7265746963616C20> Tj 1 0 0 1 399.36 168.48 Tm <70726F626162696C69747920> Tj 1 0 0 1 455.52 168.24 Tm <666F7220> Tj 1 0 0 1 472.56 168.48 Tm <67657474696E6720> Tj 1 0 0 1 509.28 169.20 Tm <227461696C7322> Tj 0.96 Tc 1 0 0 1 60.00 155.04 Tm <697320> Tj 1 0 0 1 71.04 155.04 Tm <2520> Tj 1 0 0 1 83.04 155.04 Tm <6F7220> Tj 1 0 0 1 96.00 155.04 Tm <302E352E20> Tj 1 0 0 1 119.52 155.04 Tm <696620> Tj 1 0 0 1 130.80 154.80 Tm <4920> Tj 1 0 0 1 138.00 154.80 Tm <6F6E6C7920> Tj 1 0 0 1 162.72 155.04 Tm <666C697020> Tj 1 0 0 1 182.64 155.04 Tm <74686520> Tj 1 0 0 1 200.40 154.80 Tm <636F696E20> Tj 1 0 0 1 224.16 154.56 Tm <6F6E636520> Tj 1 0 0 1 249.84 154.80 Tm <616E6420> Tj 1 0 0 1 269.76 154.80 Tm <68617070656E20> Tj 1 0 0 1 307.44 154.56 Tm <746F20> Tj 1 0 0 1 320.16 154.80 Tm <67657420> Tj 1 0 0 1 337.44 154.56 Tm <7461696C732C20> Tj 1 0 0 1 363.60 154.56 Tm <4920> Tj 1 0 0 1 370.56 154.80 Tm <776F756C6420> Tj 1 0 0 1 403.92 154.32 Tm <636F6E636C75646520> Tj 1 0 0 1 450.72 154.08 Tm <7468617420> Tj 1 0 0 1 471.84 154.08 Tm <50287461696C732920> Tj 1 0 0 1 510.00 155.52 Tm <3D20> Tj 1 0 0 1 521.04 154.08 Tm <312E> Tj 0.72 Tc 1 0 0 1 59.76 141.12 Tm <5468697320> Tj 1 0 0 1 84.24 141.12 Tm <697320> Tj 1 0 0 1 95.52 140.88 Tm <6F6276696F75736C7920> Tj 1 0 0 1 145.68 140.88 Tm <6E6F7420> Tj 1 0 0 1 164.40 140.88 Tm <636F72726563742E20> Tj 1 0 0 1 207.36 140.88 Tm <4920> Tj 1 0 0 1 214.32 140.88 Tm <6D75737420> Tj 1 0 0 1 240.48 140.64 Tm <666C697020> Tj 1 0 0 1 259.68 140.88 Tm <74686520> Tj 1 0 0 1 277.44 140.64 Tm <636F696E20> Tj 1 0 0 1 301.20 140.64 Tm <6D616E7920> Tj 1 0 0 1 330.96 140.64 Tm <6D6F726520> Tj 1 0 0 1 358.80 140.64 Tm <74696D657320> Tj 1 0 0 1 387.84 140.40 Tm <746F20> Tj 1 0 0 1 400.56 137.76 Tm <5F73657420> Tj 1 0 0 1 418.32 140.40 Tm <6120> Tj 1 0 0 1 426.72 140.40 Tm <62657474657220> Tj 1 0 0 1 457.44 140.16 Tm <617070726F78696D6174696F6E20> Tj 1 0 0 1 530.88 139.92 Tm <6F49> Tj 0.72 Tc 1 0 0 1 59.76 126.96 Tm <74686520> Tj 1 0 0 1 77.28 126.96 Tm <7468656F7265746963616C20> Tj 1 0 0 1 130.80 126.96 Tm <70726F626162696C6974792E20> Tj 1 0 0 1 193.44 126.72 Tm <4D6179626520> Tj 1 0 0 1 229.92 126.72 Tm <696620> Tj 1 0 0 1 240.24 126.72 Tm <4920> Tj 1 0 0 1 246.96 126.72 Tm <666C697020> Tj 1 0 0 1 266.64 126.48 Tm <74686520> Tj 1 0 0 1 284.16 126.48 Tm <636F696E20> Tj 1 0 0 1 309.12 126.48 Tm <31303020> Tj 1 0 0 1 329.28 126.24 Tm <74696D65732C20> Tj 1 0 0 1 361.20 126.24 Tm <4920> Tj 1 0 0 1 368.16 126.72 Tm <67657420> Tj 1 0 0 1 386.16 126.24 Tm <352D3520> Tj 1 0 0 1 401.04 126.00 Tm <74616969732E20> Tj 1 0 0 1 430.56 126.00 Tm <4920> Tj 1 0 0 1 437.76 126.24 Tm <722E766F756C6420> Tj 1 0 0 1 470.64 126.00 Tm <636F6E636C75646520> Tj 1 0 0 1 517.44 126.00 Tm <74686174> Tj 0.96 Tc 1 0 0 1 59.52 112.80 Tm <502874616969732920> Tj 1 0 0 1 97.44 114.24 Tm <3D20> Tj 1 0 0 1 107.04 112.56 Tm <302E35352C20> Tj 1 0 0 1 134.40 112.80 Tm <776869636820> Tj 1 0 0 1 167.04 112.56 Tm <697320> Tj 1 0 0 1 178.32 112.56 Tm <6120> Tj 1 0 0 1 186.72 112.56 Tm <6C6F7420> Tj 1 0 0 1 202.32 112.56 Tm <636C6F73657220> Tj 1 0 0 1 234.00 112.32 Tm <746F20> Tj 1 0 0 1 246.00 112.32 Tm <302E352E20> Tj 1 0 0 1 270.00 112.32 Tm <496620> Tj 1 0 0 1 280.80 112.32 Tm <4920> Tj 1 0 0 1 288.00 112.32 Tm <666C697020> Tj 1 0 0 1 307.68 112.32 Tm <74686520> Tj 1 0 0 1 325.44 112.32 Tm <636F726E20> Tj 1 0 0 1 350.16 112.32 Tm <31302C30303020> Tj 1 0 0 1 385.20 112.08 Tm <74696D65732E20> Tj 1 0 0 1 417.60 112.08 Tm <4920> Tj 1 0 0 1 425.04 112.32 Tm <76726F756C6420> Tj 1 0 0 1 457.92 111.84 Tm <65787065637420> Tj 1 0 0 1 492.24 111.84 Tm <726F20> Tj 1 0 0 1 505.20 109.20 Tm <5F71657220> Tj 1 0 0 1 522.96 111.84 Tm <616E> Tj 0.72 Tc 1 0 0 1 59.52 98.88 Tm <6578706572696D656E74616C20> Tj 1 0 0 1 125.28 98.88 Tm <76616C756520> Tj 1 0 0 1 155.28 99.12 Tm <7665727920> Tj 1 0 0 1 179.28 98.64 Tm <636C6F736520> Tj 1 0 0 1 207.12 98.40 Tm <746F20> Tj 1 0 0 1 219.60 98.40 Tm <74686520> Tj 1 0 0 1 237.12 98.40 Tm <7468656F7265746963616C20> Tj 1 0 0 1 290.40 98.40 Tm <76616C756520> Tj 1 0 0 1 319.68 98.16 Tm <6F6620> Tj 1 0 0 1 332.64 98.16 Tm <302E352E> Tj 0.96 Tc /F3 9.0 Tf 1 0 0 1 60.72 33.12 Tm <6920> Tj 1 0 0 1 66.96 33.12 Tm <6F6620> Tj 1 0 0 1 78.00 33.12 Tm <36> Tj 0.72 Tc 1 0 0 1 502.32 32.40 Tm <7265766973656420> Tj 1 0 0 1 534.96 32.16 Tm <6C326C3139313036> Tj ET endstream endobj 4 0 obj 17995 endobj 5 0 obj << /Type /Font /Subtype /Type1 /BaseFont /Helvetica /Encoding /WinAnsiEncoding >> endobj 6 0 obj << /Length 7 0 R >> stream ffixs&e&&fW Plobability'is a huge topic and we won't be able to cover eu".ythinffre. but \\,e can get to some hrasics.First of all. u'e ma1, classify most probabilities into one of two categones: classicalll, (theoreticallr,)determined probabi li ties and experimentally determined probabi li ties.Ciassicallv determined probabilities require that each possible event have an equal chance of occumns. P(,{) = number of ways event A mal'occur totai number of diffbrent outcomes oossible For example. let's say I want to know the probability of rolling a "4"'uvith a fair die. The notation "P(A)" means"the probability of event A." n, rr. ,, i,\ number of ways that a "4" mav occurt'l rolllnsu + l=number of diff-erent ourcomes Note that a ''4" can onlyoccur in one wav. since thereis oniy one "4" on a die.Note also that the number ofdift-erent outcomes is 6. srncea die has six sides. Expenmentaliy dctermined probabilities areobsen,e) several trials and count horv many based on many observations or tnals. We would perform (ortimes we get a desired result. P(A) = number of times event A occurs total number of tnals For example. I may find the probability" of making a free throu inbasketball by going out and attempting a free throw manv times.Then the probability rs: number of times I make a free throw Ptmakine a frcc thror,r ; = total numbcr of attemots This is reallv an approxtmation because if I go out the next da1' and do the same expenment, I may get adifferent probabiiity. It makes sense that the more attempts I try, the closer the expenmental probabilityu ill get to the actual probability. To understand this idea that more trials leads to a better approrimarion.imagine the cxperiment of flipping a coln.Because there are two sides (heads/tails) on a fair coin, the theoretical probability for getting "tails"is % or 0.5. if I only flip the coin once and happen to get tails, I would conclude that P(tails) = 1.This is obviously not correct. I must flip the coin many more times to _set a better approximation oIthe theoretical probability. Maybe if I flip the coin 100 times, I get 5-5 taiis. I r.vould conclude thatP(taiis) = 0.55, which is a lot closer to 0.5. If I flip the corn 10,000 times. 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380.64 702.96 Tm <63616E20> Tj 1 0 0 1 400.08 702.96 Tm <66696E6420> Tj 1 0 0 1 423.12 702.96 Tm <6120> Tj 1 0 0 1 431.52 702.96 Tm <7468656F7265746963616920> Tj 1 0 0 1 486.48 702.96 Tm <76616C756520> Tj 1 0 0 1 515.28 702.72 Tm <616E6420> Tj 1 0 0 1 535.44 702.72 Tm <7468656E20> Tj 1 0 0 1 558.72 702.72 Tm <747276> Tj 0.72 Tc 1 0 0 1 64.08 689.04 Tm <746F20> Tj 1 0 0 1 76.32 689.52 Tm <76656E667920> Tj 1 0 0 1 107.52 689.04 Tm <697420> Tj 1 0 0 1 117.12 689.04 Tm <6578706572696D656E74616C6C7920> Tj 1 0 0 1 192.96 689.76 Tm <28617320> Tj 1 0 0 1 210.00 689.04 Tm <696E20> Tj 1 0 0 1 222.72 689.04 Tm <6F757220> Tj 1 0 0 1 241.68 689.04 Tm <6578706572696D656E7420> Tj 1 0 0 1 298.80 688.80 Tm <6F6620> Tj 1 0 0 1 312.00 688.80 Tm <666C697070696E5F6520> Tj 1 0 0 1 353.76 689.04 Tm <6120> Tj 1 0 0 1 362.16 689.04 Tm <636F696E20> Tj 1 0 0 1 386.16 688.80 Tm <7365766572616C20> Tj 1 0 0 1 423.12 688.80 Tm <74696D6573292E> Tj 0.72 Tc 1 0 0 1 63.84 660.96 Tm <5768617420> Tj 1 0 0 1 92.64 660.72 Tm <697320> Tj 1 0 0 1 103.68 660.72 Tm <74686520> Tj 1 0 0 1 121.20 661.20 Tm <70726F626162696C69747920> Tj 1 0 0 1 177.36 660.72 Tm <6F6620> Tj 1 0 0 1 190.08 660.72 Tm <726F6C6C696E6720> Tj 1 0 0 1 225.84 660.96 Tm <6120> Tj 1 0 0 1 234.48 661.92 Tm <22372220> Tj 1 0 0 1 254.16 660.96 Tm <7769746820> Tj 1 0 0 1 278.64 660.72 Tm <6120> Tj 1 0 0 1 286.80 660.72 Tm <73696E676C6520> Tj 1 0 0 1 318.48 660.72 Tm <6661697220> Tj 1 0 0 1 338.64 660.72 Tm <6469653F20> Tj 1 0 0 1 364.80 660.72 Tm <54686520> Tj 1 0 0 1 386.88 660.72 Tm <616E7377657220> Tj 1 0 0 1 423.84 660.72 Tm <697320> Tj 1 0 0 1 435.12 660.72 Tm <7A65726F2C20> Tj 1 0 0 1 462.24 660.48 Tm <6265636175736520> Tj 1 0 0 1 503.76 660.48 Tm <746865726520> Tj 1 0 0 1 530.64 660.48 Tm <69736E277420> Tj 1 0 0 1 555.12 660.48 Tm <6120> Tj 1 0 0 1 563.28 661.68 Tm <223722> Tj 0.72 Tc 1 0 0 1 63.84 647.04 Tm <6F6E20> Tj 1 0 0 1 78.96 647.04 Tm <74686520> Tj 1 0 0 1 96.24 647.04 Tm <6469652E20> Tj 1 0 0 1 120.00 646.80 Tm <536F2C20> Tj 1 0 0 1 138.96 647.04 Tm <74686520> Tj 1 0 0 1 156.72 647.28 Tm <70726F626162696C69747920> Tj 1 0 0 1 212.88 647.04 Tm <6F6620> Tj 1 0 0 1 225.84 647.04 Tm <616E20> Tj 1 0 0 1 240.24 647.04 Tm <696D706F737369626C6520> Tj 1 0 0 1 295.20 646.80 Tm <6576656E7420> Tj 1 0 0 1 324.96 646.80 Tm <6F6363757272696E6720> Tj 1 0 0 1 374.40 647.04 Tm <697320> Tj 1 0 0 1 385.20 646.80 Tm <7A65726F2E20> Tj 1 0 0 1 415.20 647.04 Tm <5768617420> Tj 1 0 0 1 444.72 646.80 Tm <697320> Tj 1 0 0 1 456.00 646.80 Tm <74686520> Tj 1 0 0 1 474.24 647.04 Tm <70726F626162696C69747920> Tj 1 0 0 1 529.92 646.56 Tm <6F6620> Tj 1 0 0 1 542.88 646.56 Tm <726F6C6C726E67> Tj 0.72 Tc 1 0 0 1 63.84 633.36 Tm <6120> Tj 1 0 0 1 72.00 633.36 Tm <6E756D62657220> Tj 1 0 0 1 111.36 633.12 Tm <6265747765656E20> Tj 1 0 0 1 155.04 633.12 Tm <6F6E6520> Tj 1 0 0 1 175.68 633.12 Tm <616E6420> Tj 1 0 0 1 196.08 633.12 Tm <7369782C20> Tj 1 0 0 1 216.24 633.12 Tm <696E636C75736976653F20> Tj 1 0 0 1 270.96 633.12 Tm <54686520> Tj 1 0 0 1 293.04 633.12 Tm <616E7377657220> Tj 1 0 0 1 330.00 633.12 Tm <6865726520> Tj 1 0 0 1 354.00 633.12 Tm <697320> Tj 1 0 0 1 365.28 632.88 Tm <6F6E652C20> Tj 1 0 0 1 388.80 632.88 Tm <73696E636520> Tj 1 0 0 1 416.64 633.12 Tm <616C6C20> Tj 1 0 0 1 432.00 632.88 Tm <74686520> Tj 1 0 0 1 449.52 632.88 Tm <6E756D6265727320> Tj 1 0 0 1 495.12 632.64 Tm <6F6E20> Tj 1 0 0 1 510.00 632.64 Tm <6120> Tj 1 0 0 1 518.16 632.64 Tm <64696520> Tj 1 0 0 1 536.16 632.64 Tm <617265> Tj 0.72 Tc 1 0 0 1 63.84 619.44 Tm <6265747765656E20> Tj 1 0 0 1 106.32 619.44 Tm <6F6E6520> Tj 1 0 0 1 126.96 619.44 Tm <616E6420> Tj 1 0 0 1 147.60 619.20 Tm <7369782C20> Tj 1 0 0 1 167.76 619.44 Tm <696E636C75736976652E20> Tj 1 0 0 1 220.80 619.44 Tm <536F2C20> Tj 1 0 0 1 239.52 619.44 Tm <74686520> Tj 1 0 0 1 257.04 619.68 Tm <70726F626162696C69747920> Tj 1 0 0 1 312.72 619.20 Tm <6F6620> Tj 1 0 0 1 326.40 619.20 Tm <616E20> Tj 1 0 0 1 340.56 619.44 Tm <6576656E7420> Tj 1 0 0 1 369.84 619.20 Tm <7468617420> Tj 1 0 0 1 390.96 619.20 Tm <697320> Tj 1 0 0 1 402.00 619.20 Tm <6365727461696E20> Tj 1 0 0 1 438.00 619.20 Tm <746F20> Tj 1 0 0 1 450.72 618.96 Tm <6F6363757220> Tj 1 0 0 1 481.20 619.20 Tm <697320> Tj 1 0 0 1 492.48 618.96 Tm <6F6E652E20> Tj 1 0 0 1 518.64 618.96 Tm <416C6C> Tj 0.72 Tc 1 0 0 1 63.60 606.00 Tm <70726F626162696C697469657320> Tj 1 0 0 1 126.24 605.76 Tm <7468657265666F72652C20> Tj 1 0 0 1 176.40 605.76 Tm <6D75737420> Tj 1 0 0 1 203.28 605.76 Tm <6C696520> Tj 1 0 0 1 218.16 605.76 Tm <6265747765656E20> Tj 1 0 0 1 261.36 605.76 Tm <7A65726F20> Tj 1 0 0 1 285.12 605.76 Tm <616E6420> Tj 1 0 0 1 305.28 605.52 Tm <6F6E652C20> Tj 1 0 0 1 329.04 605.76 Tm <696E636C75736976652E20> Tj 1 0 0 1 381.84 605.76 Tm <576520> Tj 1 0 0 1 401.76 605.52 Tm <63616E20> Tj 1 0 0 1 421.68 605.52 Tm <73617920> Tj 1 0 0 1 440.64 605.52 Tm <7468697320> Tj 1 0 0 1 461.76 605.52 Tm <73796D626F6C6963616C6C69276173> Tj 1.50 Tc 1 0 0 1 65.28 591.36 Tm <303C502841293C312E> Tj 0.72 Tc 1 0 0 1 198.96 561.12 Tm <466F7220> Tj 1 0 0 1 218.88 561.12 Tm <66757475726520> Tj 1 0 0 1 250.80 561.12 Tm <6578616D706C65732E20> Tj 1 0 0 1 302.16 561.12 Tm <776520> Tj 1 0 0 1 319.44 561.12 Tm <77696C6C20> Tj 1 0 0 1 341.28 561.12 Tm <726566657220> Tj 1 0 0 1 367.20 561.12 Tm <746F20> Tj 1 0 0 1 379.92 561.12 Tm <6120> Tj 1 0 0 1 388.32 560.88 Tm <7374616E6461726420> Tj 1 0 0 1 432.24 560.88 Tm <6465636B20> Tj 1 0 0 1 458.16 560.88 Tm <6F6620> Tj 1 0 0 1 471.60 560.88 Tm <63617264732E20> Tj 1 0 0 1 506.40 560.64 Tm <416C6C6F7720> Tj 1 0 0 1 539.28 560.64 Tm <6D6520> Tj 1 0 0 1 556.56 560.88 Tm <746F> Tj 0.72 Tc 1 0 0 1 198.96 547.20 Tm <646566696E6520> Tj 1 0 0 1 232.08 547.44 Tm <74686520> Tj 1 0 0 1 250.08 547.20 Tm <636F6E74656E747320> Tj 1 0 0 1 293.28 546.96 Tm <6F6620> Tj 1 0 0 1 305.76 547.20 Tm <74686520> Tj 1 0 0 1 324.48 547.20 Tm <6465636B20> Tj 1 0 0 1 349.92 547.20 Tm <666F7220> Tj 1 0 0 1 367.20 547.20 Tm <74686F736520> Tj 1 0 0 1 395.76 547.44 Tm <77686F20> Tj 1 0 0 1 419.76 546.96 Tm <6861766520> Tj 1 0 0 1 445.68 546.96 Tm <6E6576657220> Tj 1 0 0 1 476.16 547.20 Tm <706C6179656420> Tj 1 0 0 1 511.20 546.72 Tm <636172647320> Tj 1 0 0 1 539.52 546.72 Tm <6265666F72652E> Tj 0.72 Tc 1 0 0 1 198.96 533.28 Tm <4576656E20> Tj 1 0 0 1 226.80 533.28 Tm <696620> Tj 1 0 0 1 237.60 533.76 Tm <796F7520> Tj 1 0 0 1 258.72 533.28 Tm <6B6E6F7720> Tj 1 0 0 1 288.00 533.28 Tm <74686520> Tj 1 0 0 1 305.52 533.28 Tm <636F6E74656E74732C20> Tj 1 0 0 1 352.32 533.28 Tm <697420> Tj 1 0 0 1 362.64 533.28 Tm <77696C6920> Tj 1 0 0 1 384.00 533.04 Tm <626520> Tj 1 0 0 1 398.40 533.04 Tm <6E69636520> Tj 1 0 0 1 421.68 533.04 Tm <746F20> Tj 1 0 0 1 434.16 533.04 Tm <6861766520> Tj 1 0 0 1 460.32 533.04 Tm <6120> Tj 1 0 0 1 468.72 533.04 Tm <7461626C6520> Tj 1 0 0 1 495.84 532.80 Tm <6F6620> Tj 1 0 0 1 508.56 532.80 Tm <696E666F726D6174696F6E20> Tj 1 0 0 1 568.08 532.80 Tm <746F> Tj 0.48 Tc 1 0 0 1 198.96 519.60 Tm <7265662D657220> Tj 1 0 0 1 224.64 519.60 Tm <746F2E> Tj 0.96 Tc 1 0 0 1 63.36 477.36 Tm <436F6C6F72> Tj 0.00 Tc /F3 42.0 Tf 1 0 0 1 537.36 440.88 Tm <76> Tj 0.00 Tc /F3 45.0 Tf 1 0 0 1 546.96 409.44 Tm <74> Tj 0.00 Tc /F3 29.0 Tf 1 0 0 1 535.92 390.96 Tm <2E6C> Tj 0.00 Tc /F3 10.0 Tf 1 0 0 1 536.88 357.12 Tm <2A> Tj 0.48 Tc 1 0 0 1 63.60 449.52 Tm <726564> Tj 1.50 Tc /F3 11.0 Tf 1 0 0 1 180.72 450.24 Tm <28322C332E312C352E362E372E20> Tj 1 0 0 1 257.76 449.28 Tm <452E392E20> Tj 1 0 0 1 282.48 449.28 Tm <31302E20> Tj 1 0 0 1 299.04 449.28 Tm <4A61636B2C20> Tj 1 0 0 1 327.12 447.12 Tm <517565656E2E20> Tj 1 0 0 1 365.04 449.28 Tm <4B696E672E20> Tj 1 0 0 1 395.04 449.28 Tm <41636529> Tj 1.44 Tc 1 0 0 1 180.72 422.40 Tm <28322E332C20> Tj 1 0 0 1 208.56 421.68 Tm <342E20> Tj 1 0 0 1 221.04 421.44 Tm <352E20> Tj 1 0 0 1 233.04 421.68 Tm <362C4920> Tj 1 0 0 1 252.00 421.20 Tm <2E382E20> Tj 1 0 0 1 269.28 421.44 Tm <392E20> Tj 1 0 0 1 282.24 421.44 Tm <31302E20> Tj 1 0 0 1 298.80 421.44 Tm <4A61636B2E20> Tj 1 0 0 1 327.12 419.28 Tm <517565656E2C20> Tj 1 0 0 1 364.80 421.44 Tm <4B696E672C20> Tj 1 0 0 1 395.04 421.44 Tm <41636529> Tj 1.20 Tc 1 0 0 1 180.72 394.56 Tm <28322C20> Tj 1 0 0 1 196.80 393.36 Tm <332E20> Tj 1 0 0 1 208.56 393.60 Tm <342E352C20> Tj 1 0 0 1 233.04 393.36 Tm <362E20> Tj 1 0 0 1 245.52 393.60 Tm <372E20> Tj 1 0 0 1 257.76 393.36 Tm <382E20> Tj 1 0 0 1 269.28 393.60 Tm <392C20> Tj 1 0 0 1 282.48 393.36 Tm <31302E20> Tj 1 0 0 1 298.80 393.36 Tm <4A61636B2E20> Tj 1 0 0 1 326.88 391.20 Tm <517565656E2E20> Tj 1 0 0 1 364.80 393.36 Tm <4B696E672E20> Tj 1 0 0 1 394.80 393.60 Tm <41636529> Tj 1.50 Tc 1 0 0 1 180.72 366.48 Tm <28322C332E342C352E362E312E382C392E20> Tj 1 0 0 1 282.24 365.76 Tm <31302E20> Tj 1 0 0 1 298.80 365.52 Tm <4A61636B2E20> Tj 1 0 0 1 326.88 363.36 Tm <517565656E2C20> Tj 1 0 0 1 364.80 365.52 Tm <4B696E5F652C20> Tj 1 0 0 1 395.04 365.52 Tm <41636529> Tj 0.72 Tc 1 0 0 1 456.24 337.44 Tm <353220> Tj 1 0 0 1 471.36 337.68 Tm <636172647320> Tj 1 0 0 1 500.16 337.44 Tm <746F74616C> Tj 0.72 Tc 1 0 0 1 63.12 310.08 Tm <4164646974696F6E20> Tj 1 0 0 1 108.24 310.08 Tm <52756965> Tj 0.72 Tc 1 0 0 1 62.64 282.00 Tm <546F20> Tj 1 0 0 1 79.20 282.00 Tm <66696E6420> Tj 1 0 0 1 101.28 282.00 Tm <70726F626162696C697469657320> Tj 1 0 0 1 165.12 281.76 Tm <6F6620> Tj 1 0 0 1 177.84 282.00 Tm <74686520> Tj 1 0 0 1 195.36 282.00 Tm <666F726D20> Tj 1 0 0 1 222.00 282.00 Tm <50284120> Tj 1 0 0 1 245.04 281.76 Tm <6F7220> Tj 1 0 0 1 258.00 281.76 Tm <42292E20> Tj 1 0 0 1 276.00 282.00 Tm <776520> Tj 1 0 0 1 292.80 281.76 Tm <6D617920> Tj 1 0 0 1 316.56 281.76 Tm <656D706C6F7920> Tj 1 0 0 1 356.16 281.76 Tm <74686520> Tj 1 0 0 1 374.40 281.76 Tm <6164646974696F6E20> Tj 1 0 0 1 416.64 281.76 Tm <72756C652E20> Tj 1 0 0 1 444.72 281.76 Tm <54686520> Tj 1 0 0 1 467.04 282.00 Tm <70726F626162696C69747920> Tj 1 0 0 1 523.20 281.28 Tm <6F6620> Tj 1 0 0 1 536.16 281.76 Tm <4120> Tj 1 0 0 1 547.68 281.52 Tm <6F7220> Tj 1 0 0 1 560.16 281.76 Tm <42> Tj 0.72 Tc 1 0 0 1 63.12 268.08 Tm <762C20> Tj 1 0 0 1 72.24 268.08 Tm <696C6C20> Tj 1 0 0 1 84.48 268.08 Tm <696E636C75646520> Tj 1 0 0 1 122.64 267.84 Tm <74686520> Tj 1 0 0 1 140.40 268.08 Tm <70726F626162696969747920> Tj 1 0 0 1 196.56 267.84 Tm <6F6620> Tj 1 0 0 1 209.76 268.08 Tm <6576656E7420> Tj 1 0 0 1 239.04 268.08 Tm <4120> Tj 1 0 0 1 251.04 267.84 Tm <6F6363757272696E672C20> Tj 1 0 0 1 302.64 267.84 Tm <6576656E7420> Tj 1 0 0 1 332.16 267.84 Tm <4220> Tj 1 0 0 1 343.20 267.84 Tm <6F6363757272696E6520> Tj 1 0 0 1 392.64 267.84 Tm <6F7220> Tj 1 0 0 1 405.84 267.84 Tm <626F746820> Tj 1 0 0 1 430.56 267.84 Tm <6576656E747320> Tj 1 0 0 1 465.12 267.84 Tm <4120> Tj 1 0 0 1 477.36 267.84 Tm <616E6420> Tj 1 0 0 1 497.76 267.60 Tm <4220> Tj 1 0 0 1 508.56 267.60 Tm <6F6363757272696E6720> Tj 1 0 0 1 557.04 267.84 Tm <6174> Tj 0.72 Tc 1 0 0 1 62.88 253.92 Tm <74686520> Tj 1 0 0 1 80.64 253.92 Tm <73616D6520> Tj 1 0 0 1 108.00 254.16 Tm <74696D6520> Tj 1 0 0 1 132.24 253.92 Tm <6F6E20> Tj 1 0 0 1 147.60 253.92 Tm <6F6E6520> Tj 1 0 0 1 168.00 254.16 Tm <746E616C2E20> Tj 1 0 0 1 196.32 253.92 Tm <54686520> Tj 1 0 0 1 218.88 254.16 Tm <6164646974696F6E20> Tj 1 0 0 1 261.12 253.92 Tm <72756C6520> Tj 1 0 0 1 282.96 253.92 Tm <7374617465733A> Tj 0.48 Tc 1 0 0 1 62.88 226.32 Tm <50284120> Tj 1 0 0 1 84.96 226.32 Tm <6F7220> Tj 1 0 0 1 97.68 226.32 Tm <422920> Tj 1 0 0 1 112.80 227.76 Tm <3D20> Tj 1 0 0 1 122.40 226.32 Tm <5028412920> Tj 1 0 0 1 149.52 226.32 Tm <2B20> Tj 1 0 0 1 159.36 226.32 Tm <5028422920> Tj 1 0 0 1 184.56 228.72 Tm <2D20> Tj 1 0 0 1 193.92 226.32 Tm <50284120> Tj 1 0 0 1 216.96 226.32 Tm <616E6420> Tj 1 0 0 1 237.12 226.32 Tm <422920> Tj 1 0 0 1 252.24 229.92 Tm <2D2D2D2D2D2D2D2D3E> Tj 0.72 Tc 1 0 0 1 296.16 224.88 Tm <576520> Tj 1 0 0 1 316.32 224.64 Tm <6D75737420> Tj 1 0 0 1 343.20 224.64 Tm <737562747261637420> Tj 1 0 0 1 384.24 224.64 Tm <74686520> Tj 1 0 0 1 402.00 224.88 Tm <70726F626162696C69747920> Tj 1 0 0 1 458.40 224.64 Tm <6F6620> Tj 1 0 0 1 472.08 224.64 Tm <4120> Tj 1 0 0 1 484.08 224.64 Tm <616E6420> Tj 1 0 0 1 504.24 224.40 Tm <4220> Tj 1 0 0 1 515.04 224.40 Tm <6F6363756D6E6720> Tj 1 0 0 1 563.76 224.40 Tm <6174> Tj 0.72 Tc 1 0 0 1 296.16 210.72 Tm <74686520> Tj 1 0 0 1 314.40 210.72 Tm <73616D6520> Tj 1 0 0 1 342.24 210.72 Tm <74696D6520> Tj 1 0 0 1 366.96 210.72 Tm <6265636175736520> Tj 1 0 0 1 408.00 210.96 Tm <776520> Tj 1 0 0 1 425.28 210.72 Tm <646F6E277420> Tj 1 0 0 1 454.08 210.96 Tm <77616E7420> Tj 1 0 0 1 481.20 210.72 Tm <746F20> Tj 1 0 0 1 493.68 210.72 Tm <636F756E7420> Tj 1 0 0 1 522.96 210.48 Tm <7468657365> Tj 0.72 Tc 1 0 0 1 296.40 196.80 Tm <6F7574636F6D657320> Tj 1 0 0 1 346.08 196.80 Tm <616761696E2E20> Tj 1 0 0 1 381.12 196.80 Tm <5468657920> Tj 1 0 0 1 409.20 197.04 Tm <7765726520> Tj 1 0 0 1 435.84 196.80 Tm <616C726561647920> Tj 1 0 0 1 474.96 196.80 Tm <696E636C7564656420> Tj 1 0 0 1 519.36 196.56 Tm <696E20> Tj 1 0 0 1 531.60 196.56 Tm <50284129> Tj 0.96 Tc 1 0 0 1 296.64 183.60 Tm <28636F756E74656420> Tj 1 0 0 1 341.76 182.88 Tm <6F6E63652920> Tj 1 0 0 1 372.00 182.64 Tm <616E6420> Tj 1 0 0 1 391.92 182.88 Tm <5028422920> Tj 1 0 0 1 418.56 183.60 Tm <28636F756E74656420> Tj 1 0 0 1 463.68 182.88 Tm <7477696365292C20> Tj 1 0 0 1 500.88 182.40 Tm <736F20> Tj 1 0 0 1 514.32 182.64 Tm <776520> Tj 1 0 0 1 530.88 182.64 Tm <6D757374> Tj 0.72 Tc 1 0 0 1 296.40 168.24 Tm <737562747261637420> Tj 1 0 0 1 337.68 168.48 Tm <50284120> Tj 1 0 0 1 360.72 168.48 Tm <616E6420> Tj 1 0 0 1 380.88 168.48 Tm <422920> Tj 1 0 0 1 395.76 168.48 Tm <746F20> Tj 1 0 0 1 408.48 168.48 Tm <61766F696420> Tj 1 0 0 1 438.48 168.24 Tm <7468697320> Tj 1 0 0 1 459.36 168.48 Tm <646F75626C652D636F756E72696E732E> Tj 0.72 Tc 1 0 0 1 62.64 128.40 Tm <4C6574277320> Tj 1 0 0 1 90.24 128.16 Tm <75736520> Tj 1 0 0 1 109.20 128.16 Tm <616E20> Tj 1 0 0 1 123.12 128.16 Tm <6578616D706C6520> Tj 1 0 0 1 167.52 128.40 Tm <66726F6D20> Tj 1 0 0 1 194.16 128.16 Tm <6F757220> Tj 1 0 0 1 213.60 128.16 Tm <7374616E6461726420> Tj 1 0 0 1 257.52 128.16 Tm <6465636B20> Tj 1 0 0 1 282.96 128.16 Tm <6F6620> Tj 1 0 0 1 295.68 128.16 Tm <63617264732E> Tj 0.72 Tc 1 0 0 1 62.64 100.56 Tm <5768617420> Tj 1 0 0 1 91.44 100.32 Tm <697320> Tj 1 0 0 1 102.48 100.32 Tm <74686520> Tj 1 0 0 1 120.00 100.56 Tm <70726F626162696C69747920> Tj 1 0 0 1 176.16 100.32 Tm <6F6620> Tj 1 0 0 1 189.12 100.08 Tm <6265696E6720> Tj 1 0 0 1 219.12 100.32 Tm <6465616C7420> Tj 1 0 0 1 246.00 100.32 Tm <6120> Tj 1 0 0 1 254.40 100.32 Tm <6B696E6720> Tj 1 0 0 1 278.64 100.08 Tm <6F7220> Tj 1 0 0 1 291.60 100.08 Tm <6120> Tj 1 0 0 1 299.76 100.08 Tm <6469616D6F6E6420> Tj 1 0 0 1 345.60 100.08 Tm <66726F6D20> Tj 1 0 0 1 372.48 100.08 Tm <6120> Tj 1 0 0 1 380.88 100.08 Tm <73687566666C656420> Tj 1 0 0 1 423.12 100.08 Tm <6465636B3F20> Tj 1 0 0 1 457.68 100.08 Tm <496620> Tj 1 0 0 1 469.20 100.08 Tm <776520> Tj 1 0 0 1 486.72 99.84 Tm <6170706C7920> Tj 1 0 0 1 516.00 99.84 Tm <74686520> Tj 1 0 0 1 533.76 99.84 Tm <6164646974696F6E> Tj 0.72 Tc 1 0 0 1 62.64 86.64 Tm <72756C6520> Tj 1 0 0 1 84.00 86.64 Tm <746F20> 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some probabilities, it is rather easy to find a theoretical value (like the probability of getting "tails"when flipping a coin). Other probabilities lend themselves to the experimentai approach and it is difficultand very complicated to find a theoretical value. Sometimes. one can find a theoreticai value and then trvto venfy it experimentally (as in our experiment of flippin_e a coin several times).What is the probability of rolling a "7" with a single fair die? The answer is zero, because there isn't a "7"on the die. So, the probability of an impossible event occurring is zero. What is the probability of rollrnga number between one and six, inclusive? The answer here is one, since all the numbers on a die arebetween one and six, inclusive. So, the probability of an event that is certain to occur is one. Allprobabilities therefore, must lie between zero and one, inclusive. We can say this symbolicalli'as0 We must subtract the probability of A and B occumng atthe same time because we don't want to count theseoutcomes again. They were already included in P(A)(counted once) and P(B) (counted twice), so we mustsubtract P(A and B) to avoid this double-counrins. Let's use an example from our standard deck of cards.What is the probability of being dealt a king or a diamond from a shuffled deck? 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<616C726561647920> Tj 1 0 0 1 276.00 53.76 Tm <6F636375727265632E> Tj 0.72 Tc /F3 9.0 Tf 1 0 0 1 503.52 30.72 Tm <7265766973656420> Tj 1 0 0 1 536.40 30.48 Tm <6C326C3139313036> Tj ET endstream endobj 23 0 obj 16936 endobj 24 0 obj << /Length 25 0 R >> stream P(king or diamond) = P(king) + P(diamond) - P(king and diamond) T There are z[ kingsin the deck and 52total cards. P(A and B) = P(A) * P(BlA)3 of 6 There are i3diamonds and 52total cards. There is one kingof diamonds and5l total cards. P(H or S) = P(H) + P(S) - P(H & S) \-.z--?->'\\-:- 13 13 0 26 I ,<-:-r\'''l525252522 Note that there aren't any cards that are bothhearts and spades at the same time. Twoevents that cannot occur at the same time arcsaid to be mutuallv exclusive. Multiplication RuleThc muitiplication rule applies to repeated experiments or tnals. Thrs mcans that we are no longer beingdealt oniy one card, but rather ser,erai cards. Before we look at some examples, we need to discussindependent vs. dependent events. Tr.vo events are said to be independent if the occurrence of one eventdoes not have an etfect on the likelihood of the other event occumng. Trvo dependent events are events,"vhere the occurrence of one does influence the probability of the other occurring.As an example. consider the expenment of being dealt a couple cards from our deck. If I put the first cardI'm dealt back into the deck before being dealt the second card (I'm replacin_e the first card back into thedeck), then the two tnals are independent (I am randomly selecting from 52 cards rn each trial). However.if I do not replace the first card before being dealt the second. then the two trials are dependent (in thesecond trial. I only have 51 cards left to pick from, since the one selected on the first trial is not replaced).Remember. in the context of the multiplication rule, with replacement means independent and withoutreplacement means dependent.So tor independent events, the multiplication rule is P(A and B) = P(A) * P(B) Note here that P(A and B) ref-ers to the probability of A on one trial andB on the next trial, not the probability of A and B occumngsimultaneously on one trial, which rs what this notation referred to withresDect to the addition rule. And for dependent events. the multiplication rule is P(BlA) means the probability of event B occurring. given that event Ahas already occurrec. revised l2l19106 endstream endobj 25 0 obj 2136 endobj 26 0 obj << /Length 27 >> stream % CANON_PFINF_TYPE0_TEXTON endstream endobj 27 0 obj << /Length 45 >> stream q 612.24 0 0 791.76 0.00 0.00 cm /Obj21 Do Q endstream endobj 28 0 obj << /Type /Page /MediaBox [ 0 0 612.24 791.76 ] /Parent 11 0 R /Resources << /Font << /F3 5 0 R >> /XObject << /Obj21 21 0 R >> /Subtype 24 0 R /ProcSet [ /PDF /Text /ImageB /ImageC /ImageI ] >> /Contents [ 26 0 R 27 0 R 22 0 R ] >> endobj 29 0 obj << /Type /XObject /Subtype /Image /Width 2551 /Height 3299 /BitsPerComponent 1 /ColorSpace /DeviceGray /Filter /CCITTFaxDecode /DecodeParms << /K -1 /Columns 2551 >> /Length 49936 >> stream <Ç "E&ɱai1G|ԇ+3A֭C>in]8h{A_Adg|I.ʒR[_uIZ] 4α+JɄJ#!<&_ ~LhVFⅹF |bj$B\+)3 + d " ^H":D.N"(2P g# ' Lv@ 2AA@v0F p0. 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602.88 Tm <74776F20> Tj 1 0 0 1 265.44 602.88 Tm <68656166697320> Tj 1 0 0 1 297.36 603.12 Tm <776974686F757420> Tj 1 0 0 1 337.44 603.12 Tm <7265706C6163656D656E743F> Tj 0.00 Tc /F3 13.0 Tf 1 0 0 1 175.92 566.64 Tm <48494849> Tj 0.48 Tc /F3 12.0 Tf 1 0 0 1 300.24 574.32 Tm <31353620> Tj 1 0 0 1 356.16 574.32 Tm <4920> Tj 1 0 0 1 387.60 573.60 Tm <2C3B2D3A3E2E> Tj 1.44 Tc /F3 10.0 Tf 1 0 0 1 333.60 568.56 Tm <3D20> Tj 1 0 0 1 376.32 568.32 Tm <3D20> Tj 1 0 0 1 387.12 567.84 Tm <28286C2E7529353529> Tj 0.48 Tc /F3 12.0 Tf 1 0 0 1 297.60 557.04 Tm <3236353220> Tj 1 0 0 1 352.32 557.04 Tm <493720> Tj 1 0 0 1 391.44 564.72 Tm <2D2D2D272F> Tj 0.00 Tc /F3 10.0 Tf 1 0 0 1 285.36 528.48 Tm <5C> Tj 1.50 Tc /F3 12.0 Tf 1 0 0 1 222.00 574.32 Tm <313320> Tj 1 0 0 1 254.40 574.32 Tm <3132> Tj 1.50 Tc 1 0 0 1 221.04 556.80 Tm <353220> Tj 1 0 0 1 254.40 556.80 Tm <3531> Tj 0.72 Tc /F3 11.0 Tf 1 0 0 1 300.48 509.28 Tm <496620> Tj 1 0 0 1 311.76 509.28 Tm <6F6E6520> Tj 1 0 0 1 332.40 509.52 Tm <686561727420> Tj 1 0 0 1 359.04 509.52 Tm <68617320> Tj 1 0 0 1 378.48 509.28 Tm <6265656E20> Tj 1 0 0 1 404.64 509.52 Tm <72656D6F76656420> Tj 1 0 0 1 450.00 509.28 Tm <66726F6D> Tj 0.72 Tc 1 0 0 1 300.48 495.12 Tm <74686520> Tj 1 0 0 1 318.48 495.36 Tm <6465636B20> Tj 1 0 0 1 343.92 495.36 Tm <666F7220> Tj 1 0 0 1 360.96 495.12 Tm <74686520> Tj 1 0 0 1 379.44 495.12 Tm <7365636F6E6420> Tj 1 0 0 1 415.92 495.36 Tm <747269616C2C20> Tj 1 0 0 1 441.84 495.12 Tm <7468656E> Tj 0.72 Tc 1 0 0 1 300.48 481.20 Tm <746865726520> Tj 1 0 0 1 328.08 481.20 Tm <61726520> Tj 1 0 0 1 345.12 480.96 Tm <6F6E6C7920> Tj 1 0 0 1 371.28 481.20 Tm <313220> Tj 1 0 0 1 385.68 481.20 Tm <68656172747320> Tj 1 0 0 1 417.84 481.20 Tm <6C65667420> Tj 1 0 0 1 437.04 480.96 Tm <616E6420> Tj 1 0 0 1 457.20 480.96 Tm <3531> Tj 0.72 Tc 1 0 0 1 300.24 467.04 Tm <636172647320> Tj 1 0 0 1 329.28 467.04 Tm <6C6566742E> Tj 1.20 Tc 1 0 0 1 250.56 274.80 Tm <2D2E> Tj 1.50 Tc 1 0 0 1 210.72 263.76 Tm <2B2A2D5B273D20> Tj 1 0 0 1 274.08 264.00 Tm <313620> Tj 1 0 0 1 299.52 258.00 Tm <3D20> Tj 1 0 0 1 320.40 264.00 Tm <3420> Tj 1 0 0 1 356.64 270.72 Tm <49> Tj 0.96 Tc 1 0 0 1 207.84 246.48 Tm <353220> Tj 1 0 0 1 232.56 246.48 Tm <353120> Tj 1 0 0 1 267.12 246.72 Tm <3236353220> Tj 1 0 0 1 315.60 246.72 Tm <363633> Tj 0.72 Tc 1 0 0 1 372.96 252.72 Tm <546865726520> Tj 1 0 0 1 404.88 252.72 Tm <61726520> Tj 1 0 0 1 422.88 252.48 Tm <7374696C6C20> Tj 1 0 0 1 443.76 252.72 Tm <666F757220> Tj 1 0 0 1 467.28 252.72 Tm <37277320> Tj 1 0 0 1 485.04 252.72 Tm <696E20> Tj 1 0 0 1 497.76 252.72 Tm <746865> Tj 0.72 Tc 1 0 0 1 373.44 238.56 Tm <6465636B2E20> Tj 1 0 0 1 402.48 238.56 Tm <62757420> Tj 1 0 0 1 421.20 238.56 Tm <6F6E6C7920> Tj 1 0 0 1 445.68 238.32 Tm <353120> Tj 1 0 0 1 460.56 238.56 Tm <636172647320> Tj 1 0 0 1 489.60 238.56 Tm <6C65667420> Tj 1 0 0 1 508.80 238.56 Tm <746F74616C2E> Tj 1.50 Tc /F3 23.0 Tf 1 0 0 1 465.84 83.52 Tm <3D40> Tj 1.50 Tc /F3 12.0 Tf 1 0 0 1 273.84 190.32 Tm <3136> Tj 1.44 Tc 1 0 0 1 61.68 81.60 Tm <50282236616E6420> Tj 1 0 0 1 107.28 81.60 Tm <372220> Tj 1 0 0 1 124.08 81.36 Tm <6F72226C616E6436222920> Tj 1 0 0 1 190.56 83.04 Tm <3D20> Tj 1 0 0 1 203.52 81.60 Tm <502836616E64372920> Tj 1 0 0 1 259.44 81.84 Tm <2B20> Tj 1 0 0 1 275.28 81.84 Tm <502837616E6420> Tj 1 0 0 1 316.32 83.52 Tm <7120> Tj 1 0 0 1 332.16 83.28 Tm <3D20> Tj 1 0 0 1 343.20 72.00 Tm <2320> Tj 1 0 0 1 370.80 81.84 Tm <2E20> Tj 1 0 0 1 384.96 72.00 Tm <2A20> Tj 1 0 0 1 416.40 83.52 Tm <3D20> Tj 1 0 0 1 434.40 72.00 Tm <2A> Tj 0.72 Tc /F3 9.0 Tf 1 0 0 1 61.68 30.96 Tm <3420> Tj 1 0 0 1 69.36 30.72 Tm <6F6620> Tj 1 0 0 1 80.40 30.72 Tm <36> Tj 0.72 Tc /F3 10.0 Tf 1 0 0 1 61.68 566.40 Tm <50284820> Tj 1 0 0 1 84.48 566.40 Tm <616E6420> Tj 1 0 0 1 104.40 566.40 Tm <482920> Tj 1 0 0 1 119.76 567.84 Tm <3D20> Tj 1 0 0 1 129.36 566.40 Tm <5028482920> Tj 1 0 0 1 156.72 570.00 Tm <2A20> Tj 1 0 0 1 165.36 566.40 Tm <5028> Tj 0.00 Tc 1 0 0 1 153.84 531.36 Tm <2F> Tj 0.72 Tc /F3 11.0 Tf 1 0 0 1 74.64 509.28 Tm <54686520> Tj 1 0 0 1 96.48 509.28 Tm <6E6F746174696F6E20> Tj 1 0 0 1 138.72 509.28 Tm <6D65616E7320> Tj 1 0 0 1 172.32 509.28 Tm <74686520> Tj 1 0 0 1 190.08 509.52 Tm <70726F626162696C69747920> Tj 1 0 0 1 246.00 509.28 Tm <6F66> Tj 0.72 Tc 1 0 0 1 74.88 494.88 Tm <6265696E6720> Tj 1 0 0 1 104.40 494.88 Tm <6465616C7420> Tj 1 0 0 1 130.56 495.12 Tm <6120> Tj 1 0 0 1 138.72 495.12 Tm <686561727420> Tj 1 0 0 1 166.08 495.12 Tm <6F6E20> Tj 1 0 0 1 180.96 495.12 Tm <74686520> Tj 1 0 0 1 199.20 495.12 Tm <7365636F6E6420> Tj 1 0 0 1 234.96 495.12 Tm <747269616C2C> Tj 0.72 Tc 1 0 0 1 74.88 480.24 Tm <736976656E20> Tj 1 0 0 1 104.40 480.96 Tm <7468617420> Tj 1 0 0 1 125.28 481.20 Tm <796F7527766520> Tj 1 0 0 1 162.00 480.96 Tm <616C726561647920> Tj 1 0 0 1 200.16 480.72 Tm <6265656E20> Tj 1 0 0 1 225.60 480.96 Tm <6465616C7420> Tj 1 0 0 1 252.48 480.96 Tm <61> Tj 0.48 Tc 1 0 0 1 74.88 466.80 Tm <686561727420> Tj 1 0 0 1 101.76 466.80 Tm <6F6E20> Tj 1 0 0 1 116.40 466.80 Tm <74686520> Tj 1 0 0 1 133.92 466.80 Tm <666972737420> Tj 1 0 0 1 156.72 466.80 Tm <747269616C2E> Tj 0.72 Tc 1 0 0 1 61.68 431.52 Tm <5768617420> Tj 1 0 0 1 91.20 431.52 Tm <697320> Tj 1 0 0 1 102.00 431.52 Tm <74686520> Tj 1 0 0 1 119.04 431.76 Tm <70726F626162696C69747920> Tj 1 0 0 1 174.96 431.52 Tm <6F6620> Tj 1 0 0 1 188.16 431.52 Tm <6265696E6720> Tj 1 0 0 1 217.68 431.52 Tm <6465616C7420> Tj 1 0 0 1 244.32 431.76 Tm <6120> Tj 1 0 0 1 252.72 432.72 Tm <22362220> Tj 1 0 0 1 272.64 431.76 Tm <616E6420> Tj 1 0 0 1 293.04 431.76 Tm <61224A27312220> Tj 1 0 0 1 330.00 431.76 Tm <4E6F746520> Tj 1 0 0 1 355.92 431.76 Tm <7468617420> Tj 1 0 0 1 377.52 431.76 Tm <7468697320> Tj 1 0 0 1 398.40 432.00 Tm <70726F626C656D20> Tj 1 0 0 1 441.84 431.76 Tm <646F657320> Tj 1 0 0 1 466.80 431.76 Tm <6E6F7420> Tj 1 0 0 1 486.00 431.76 Tm <73612920> Tj 1 0 0 1 505.20 431.76 Tm <696E20> Tj 1 0 0 1 517.68 432.00 Tm <75276869636820> Tj 1 0 0 1 549.84 431.76 Tm <6F72646572> Tj 0.72 Tc 1 0 0 1 61.68 417.84 Tm <796F7520> Tj 1 0 0 1 83.04 417.84 Tm <61726520> Tj 1 0 0 1 100.56 417.60 Tm <6465616C7420> Tj 1 0 0 1 126.48 417.60 Tm <746865736520> Tj 1 0 0 1 154.32 417.84 Tm <74776F20> Tj 1 0 0 1 175.20 417.84 Tm <63617264732C20> Tj 1 0 0 1 206.88 417.84 Tm <6E6F7220> Tj 1 0 0 1 225.60 417.84 Tm <646F657320> Tj 1 0 0 1 250.80 417.84 Tm <697420> Tj 1 0 0 1 260.88 417.84 Tm <73617920> Tj 1 0 0 1 279.84 417.84 Tm <7769746820> Tj 1 0 0 1 304.56 417.84 Tm <6F7220> Tj 1 0 0 1 318.00 418.08 Tm <776974686F757420> Tj 1 0 0 1 357.36 418.08 Tm <7265706C6163656D656E742E20> Tj 1 0 0 1 426.48 418.08 Tm <576520> Tj 1 0 0 1 446.40 417.84 Tm <6D75737420> Tj 1 0 0 1 473.28 417.84 Tm <75736520> Tj 1 0 0 1 492.72 417.84 Tm <6120> Tj 1 0 0 1 501.36 418.08 Tm <6C6974746C6520> Tj 1 0 0 1 525.84 417.84 Tm <636F6D6D6F6E> Tj 0.96 Tc 1 0 0 1 61.92 403.68 Tm <73656E736520> Tj 1 0 0 1 90.96 403.92 Tm <746F20> Tj 1 0 0 1 103.20 403.92 Tm <64657465726D696E6520> Tj 1 0 0 1 154.08 403.92 Tm <746865736520> Tj 1 0 0 1 181.92 403.92 Tm <7468696E67732E20> Tj 1 0 0 1 220.08 403.92 Tm <5768656E20> Tj 1 0 0 1 252.00 404.16 Tm <70696179696E6720> Tj 1 0 0 1 291.12 403.92 Tm <63617264732E20> Tj 1 0 0 1 323.28 403.92 Tm <6F6E6520> Tj 1 0 0 1 343.20 404.16 Tm <726172656C7920> Tj 1 0 0 1 374.64 404.16 Tm <7075747320> Tj 1 0 0 1 398.64 404.16 Tm <6120> Tj 1 0 0 1 406.56 404.16 Tm <6361726420> Tj 1 0 0 1 431.04 403.92 Tm <7468657620> Tj 1 0 0 1 454.80 404.16 Tm <61726520> Tj 1 0 0 1 472.32 404.16 Tm <6465616C7420> Tj 1 0 0 1 499.68 404.88 Tm <6861636B20> Tj 1 0 0 1 525.12 404.16 Tm <696E746F20> Tj 1 0 0 1 546.72 404.16 Tm <746865> Tj 0.72 Tc 1 0 0 1 61.68 389.76 Tm <6465636B20> Tj 1 0 0 1 87.60 389.76 Tm <6265666F726520> Tj 1 0 0 1 120.48 389.76 Tm <6265696E6720> Tj 1 0 0 1 150.72 389.76 Tm <6465616C7420> Tj 1 0 0 1 177.12 390.00 Tm <74686520> Tj 1 0 0 1 194.88 390.00 Tm <6E65787420> Tj 1 0 0 1 218.64 389.76 Tm <636172642C20> Tj 1 0 0 1 245.52 390.00 Tm <736F20> Tj 1 0 0 1 259.20 390.00 Tm <776520> Tj 1 0 0 1 276.24 390.24 Tm <77696C6C20> Tj 1 0 0 1 297.84 390.00 Tm <696E7465727072657420> Tj 1 0 0 1 342.00 390.00 Tm <7468697320> Tj 1 0 0 1 362.40 390.24 Tm <70726F626C656D20> Tj 1 0 0 1 406.56 390.00 Tm <617320> Tj 1 0 0 1 419.76 390.00 Tm <6120> Tj 1 0 0 1 428.40 390.96 Tm <22776974686F757420> Tj 1 0 0 1 473.52 390.24 Tm <7265706C6163656D656E7422> Tj 0.72 Tc 1 0 0 1 61.68 376.08 Tm <70726F626C656D2E20> Tj 1 0 0 1 110.64 375.84 Tm <416C736F2E20> Tj 1 0 0 1 139.44 375.84 Tm <73696E636520> Tj 1 0 0 1 167.28 375.84 Tm <616E20> Tj 1 0 0 1 181.44 375.84 Tm <6F7264657220> Tj 1 0 0 1 209.76 375.84 Tm <697320> Tj 1 0 0 1 220.80 375.84 Tm <6E6F7420> Tj 1 0 0 1 239.28 376.32 Tm <676976656E20> Tj 1 0 0 1 269.04 375.84 Tm <666F7220> Tj 1 0 0 1 286.08 376.08 Tm <74686520> Tj 1 0 0 1 304.08 376.08 Tm <7477276F20> Tj 1 0 0 1 325.44 376.08 Tm <63617264732C20> Tj 1 0 0 1 356.64 376.32 Tm <776520> Tj 1 0 0 1 374.40 376.08 Tm <77696C6C20> Tj 1 0 0 1 396.24 376.08 Tm <6861766520> Tj 1 0 0 1 422.16 376.08 Tm <746F20> Tj 1 0 0 1 434.88 375.84 Tm <636F6E736964657220> Tj 1 0 0 1 478.80 376.08 Tm <74686520> Tj 1 0 0 1 497.04 376.32 Tm <706F73737262696C69747920> Tj 1 0 0 1 550.08 376.08 Tm <6F66> Tj 0.72 Tc 1 0 0 1 61.92 361.68 Tm <6265696E6720> Tj 1 0 0 1 91.68 361.92 Tm <6465616C7420> Tj 1 0 0 1 117.60 361.92 Tm <6120> Tj 1 0 0 1 125.76 363.60 Tm <22362220> Tj 1 0 0 1 145.92 361.92 Tm <616E6420> Tj 1 0 0 1 166.08 362.16 Tm <7468656E20> Tj 1 0 0 1 190.08 361.92 Tm <6120> Tj 1 0 0 1 198.48 363.60 Tm <224A2220> Tj 1 0 0 1 217.68 361.92 Tm <6F7220> Tj 1 0 0 1 230.88 362.16 Tm <6120> Tj 1 0 0 1 239.28 362.88 Tm <22372220> Tj 1 0 0 1 258.96 361.92 Tm <616E6420> Tj 1 0 0 1 279.36 362.16 Tm <7468656E20> Tj 1 0 0 1 303.60 362.16 Tm <6120> Tj 1 0 0 1 312.00 363.84 Tm <22362E2220> Tj 1 0 0 1 334.56 362.16 Tm <426F746820> Tj 1 0 0 1 360.96 361.92 Tm <6F6620> Tj 1 0 0 1 374.16 361.92 Tm <746865736520> Tj 1 0 0 1 402.72 362.16 Tm <736974756174696F6E7320> Tj 1 0 0 1 452.16 362.16 Tm <61726520> Tj 1 0 0 1 469.92 362.16 Tm <696D706C69656420> Tj 1 0 0 1 510.24 362.16 Tm <696E20> Tj 1 0 0 1 522.72 362.16 Tm <746865> Tj 0.72 Tc 1 0 0 1 61.68 348.00 Tm <7175657374696F6E2C20> Tj 1 0 0 1 108.48 347.76 Tm <6576656E20> Tj 1 0 0 1 133.68 347.76 Tm <74686F75676820> Tj 1 0 0 1 170.16 348.00 Tm <7468657920> Tj 1 0 0 1 194.40 348.00 Tm <61726520> Tj 1 0 0 1 211.68 348.00 Tm <6E6F7420> Tj 1 0 0 1 230.16 348.00 Tm <73746174656420> Tj 1 0 0 1 261.36 348.00 Tm <6578706C696369746C792E20> Tj 1 0 0 1 316.32 348.00 Tm <4F6674656E20> Tj 1 0 0 1 346.56 348.24 Tm <70726F626C656D7320> Tj 1 0 0 1 395.28 348.24 Tm <77696C6C20> Tj 1 0 0 1 417.36 348.00 Tm <626520> Tj 1 0 0 1 432.00 348.24 Tm <776F7264656420> Tj 1 0 0 1 470.88 348.00 Tm <696E20> Tj 1 0 0 1 483.36 348.00 Tm <7468697320> Tj 1 0 0 1 504.24 348.00 Tm <66617368696F6E2E20> Tj 1 0 0 1 546.00 348.00 Tm <616E64> Tj 0.72 Tc 1 0 0 1 61.68 334.08 Tm <72766520> Tj 1 0 0 1 78.72 333.84 Tm <6E65656420> Tj 1 0 0 1 104.40 333.84 Tm <746F20> Tj 1 0 0 1 116.16 333.84 Tm <7265616C697A6520> Tj 1 0 0 1 151.92 334.08 Tm <776869636820> Tj 1 0 0 1 184.32 334.08 Tm <6F7574636F6D657320> Tj 1 0 0 1 233.04 334.08 Tm <6E65656420> Tj 1 0 0 1 258.96 334.08 Tm <746F20> Tj 1 0 0 1 271.44 334.08 Tm <626520> Tj 1 0 0 1 285.84 334.08 Tm <696E636C7564656420> Tj 1 0 0 1 330.72 334.08 Tm <696E20> Tj 1 0 0 1 342.96 334.08 Tm <6F757220> Tj 1 0 0 1 361.92 334.08 Tm <63616C63756C6174696F6E732E20> Tj 1 0 0 1 429.84 334.08 Tm <4B65657020> Tj 1 0 0 1 458.16 334.32 Tm <7468697320> Tj 1 0 0 1 478.80 334.32 Tm <696E20> Tj 1 0 0 1 491.52 334.32 Tm <6D696E6420> Tj 1 0 0 1 519.60 334.32 Tm <7768656E> Tj 0.72 Tc 1 0 0 1 61.68 319.92 Tm <696E74657270726574726E6720> Tj 1 0 0 1 120.48 319.92 Tm <6120> Tj 1 0 0 1 128.64 320.16 Tm <70726F626C656D2E> Tj 0.72 Tc 1 0 0 1 61.44 291.84 Tm <546F20> Tj 1 0 0 1 78.48 291.84 Tm <736F6C766520> Tj 1 0 0 1 106.56 291.84 Tm <7468697320> Tj 1 0 0 1 126.48 292.32 Tm <70726F626C656D2C20> Tj 1 0 0 1 172.80 292.08 Tm <6C6574277320> Tj 1 0 0 1 196.80 292.08 Tm <636F6E736964657220> Tj 1 0 0 1 240.24 292.08 Tm <666972737420> Tj 1 0 0 1 262.56 292.08 Tm <74686520> Tj 1 0 0 1 280.32 292.32 Tm <70726F626162696C69747920> Tj 1 0 0 1 336.72 292.32 Tm <6F6620> Tj 1 0 0 1 349.44 292.32 Tm <6265696E6720> Tj 1 0 0 1 379.92 292.08 Tm <6465616C7420> Tj 1 0 0 1 406.80 292.32 Tm <6120> Tj 1 0 0 1 415.20 292.08 Tm <3620> Tj 1 0 0 1 424.56 292.08 Tm <616E6420> Tj 1 0 0 1 444.72 292.08 Tm <7468656E20> Tj 1 0 0 1 468.96 292.32 Tm <6120> Tj 1 0 0 1 477.12 292.32 Tm <373A> Tj 0.96 Tc 1 0 0 1 61.68 255.84 Tm <50283620> Tj 1 0 0 1 81.84 256.08 Tm <616E6420> Tj 1 0 0 1 102.00 256.08 Tm <372920> Tj 1 0 0 1 117.36 257.52 Tm <3D20> Tj 1 0 0 1 130.08 256.08 Tm <5028362920> Tj 1 0 0 1 154.56 259.44 Tm <2A20> Tj 1 0 0 1 163.20 256.08 Tm <50283731362920> Tj 1 0 0 1 195.36 257.52 Tm <3D> Tj 0.72 Tc 1 0 0 1 61.92 218.88 Tm <416E6420> Tj 1 0 0 1 85.44 218.88 Tm <6E6F7720> Tj 1 0 0 1 108.96 218.88 Tm <74686520> Tj 1 0 0 1 126.24 219.12 Tm <70726F626162696C69747920> Tj 1 0 0 1 182.64 218.88 Tm <6F6620> Tj 1 0 0 1 195.84 219.12 Tm <616C20> Tj 1 0 0 1 214.08 219.12 Tm <616E6420> Tj 1 0 0 1 232.80 219.12 Tm <7468656E20> Tj 1 0 0 1 257.28 219.12 Tm <6120> Tj 1 0 0 1 265.44 218.88 Tm <363A> Tj 0.72 Tc 1 0 0 1 61.68 182.64 Tm <50283720> Tj 1 0 0 1 81.60 182.64 Tm <616E6420> Tj 1 0 0 1 102.00 182.40 Tm <362920> Tj 1 0 0 1 117.36 184.08 Tm <3D20> Tj 1 0 0 1 130.08 182.40 Tm <5028372920> Tj 1 0 0 1 154.80 186.96 Tm <274620> Tj 1 0 0 1 163.20 182.40 Tm <50283631372920> Tj 1 0 0 1 195.60 184.08 Tm <3D> Tj 0.96 Tc /F3 12.0 Tf 1 0 0 1 207.84 173.04 Tm <353220> Tj 1 0 0 1 232.56 173.04 Tm <353120> Tj 1 0 0 1 267.12 173.04 Tm <3236353220> Tj 1 0 0 1 315.60 173.04 Tm <363633> Tj 0.72 Tc /F3 11.0 Tf 1 0 0 1 61.68 145.68 Tm <4E6F7720> Tj 1 0 0 1 88.32 145.92 Tm <776520> Tj 1 0 0 1 105.12 145.68 Tm <6D75737420> Tj 1 0 0 1 131.28 145.68 Tm <61646420> Tj 1 0 0 1 151.68 145.68 Tm <746865736520> Tj 1 0 0 1 179.28 145.68 Tm <74776F20> Tj 1 0 0 1 200.40 145.92 Tm <70726F626162696C697469657320> Tj 1 0 0 1 263.76 145.92 Tm <746F6765746865722E20> Tj 1 0 0 1 312.72 145.92 Tm <52656D656D62657220> Tj 1 0 0 1 368.64 145.92 Tm <74686520> Tj 1 0 0 1 387.12 145.92 Tm <6164646974696F6E20> Tj 1 0 0 1 429.84 145.68 Tm <72756C652E20> Tj 1 0 0 1 457.68 145.92 Tm <4120> Tj 1 0 0 1 470.64 143.76 Tm <7B223620> Tj 1 0 0 1 490.08 145.92 Tm <616E6420> Tj 1 0 0 1 510.48 145.92 Tm <7468656E20> Tj 1 0 0 1 534.48 145.92 Tm <6120> Tj 1 0 0 1 542.64 145.92 Tm <37227D20> Tj 1 0 0 1 562.32 145.68 Tm <6F72> Tj 0.72 Tc 1 0 0 1 61.92 131.52 Tm <6120> Tj 1 0 0 1 71.04 129.60 Tm <7B223720> Tj 1 0 0 1 91.44 131.76 Tm <616E6420> Tj 1 0 0 1 110.16 131.76 Tm <7468656E20> Tj 1 0 0 1 133.92 131.76 Tm <6120> Tj 1 0 0 1 142.32 131.52 Tm <36226920> Tj 1 0 0 1 162.72 131.76 Tm <77696C6C20> Tj 1 0 0 1 184.56 131.76 Tm <7361746973662C7620> Tj 1 0 0 1 218.64 131.76 Tm <74686520> Tj 1 0 0 1 236.16 131.76 Tm <7465726D7320> Tj 1 0 0 1 266.40 131.76 Tm <6F6620> Tj 1 0 0 1 279.36 131.76 Tm 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<746865736520> Tj 1 0 0 1 368.40 117.84 Tm <745C2C72276F20> Tj 1 0 0 1 390.24 117.84 Tm <6D75737420> Tj 1 0 0 1 416.64 117.84 Tm <626520> Tj 1 0 0 1 431.28 117.84 Tm <6D757475616C6C7920> Tj 1 0 0 1 477.36 117.84 Tm <6578636C75736976652E> Tj 0.72 Tc /F3 10.0 Tf 1 0 0 1 506.16 30.96 Tm <7265766973656420> Tj 1 0 0 1 538.80 30.96 Tm <3132313139313036> Tj ET endstream endobj 31 0 obj 18515 endobj 32 0 obj << /Length 33 0 R >> stream Let's try some exampies using the multiplication rule.What is the probability of being dealt a "6" and a"7" (tn that order) with replacement?Remember that "with replacement" means independent events.p(6and7)=p(6)*P(7)--!.+ = 16 = -l- =<60;-D52 52 2',704 169 -'--/What is the probability of being dealt two heafis without replacement? HIHI 156 I ,;-:>.= = ((l.u)55)2652 I7 ---'/ \ 13 1252 51 If one heart has been removed fromthe deck for the second trial, thenthere are only 12 hearts left and 51cards left. -.+*-['= 16 = 4 I52 51 2652 663 There are still four 7's in thedeck. but only 51 cards left total. =@ 16 P("6and 7" or"land6") = P(6and7) + P(7and q = # . * = *4 of 6 P(H and H) = P(H) * P(/ The notation means the probability ofbeing dealt a heart on the second trial,siven that you've already been dealt aheart on the first trial. What is the probability of being dealt a "6" and a"J'1" Note that this problem does not sa) in u'hich orderyou are dealt these two cards, nor does it say with or without replacement. We must use a little commonsense to determine these things. When piaying cards. one rarely puts a card thev are dealt hack into thedeck before being dealt the next card, so we will interpret this problem as a "without replacement"problem. Also. since an order is not given for the tw'o cards, we will have to consider the possrbility ofbeing dealt a "6" and then a "J" or a "7" and then a "6." Both of these situations are implied in thequestion, even though they are not stated explicitly. Often problems will be worded in this fashion. andrve need to realize which outcomes need to be included in our calculations. Keep this in mind wheninterpretrng a problem.To solve this problem, let's consider first the probability of being dealt a 6 and then a 7: P(6 and 7) = P(6) * P(716) = And now the probability of al and then a 6: P(7 and 6) = P(7) 'F P(617) = 52 51 2652 663 Now we must add these two probabilities together. Remember the addition rule. 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What is the probability of being dealt five cards and all of them being thesame surt?This problem will emplov the multiplcation rule since we are talking about repeated trials. Keep in mindthat the problem does not specify which suit the cards have to be. the only requirement is that they are allthe same suit. What this suggests is that the first card may be any suit. Then the subsequent four cardsmust match the suit that the first card "decided." The probability of the first card being any suit is one,since all the cards in the deck have a suit. P(5 cards with same suit.) - l*!*ll*lo*2=(iloo'ei)51 50 49 -+8 ------/ After the first card "decides" what all the rest have ro match. thereare i2 cards left of that suit and 51 total cards. The rest of the tnalsare also dependent on the outcomes in previous trials. We could have considered a similar case. w'here the suit was specified. For example, we could have askedforthe probability of being dealt five clubs from ourdeck. The only thing that rvouldchange in thisproblem is that the first probability would be 13152, since this first draw no longer gets ro "decide" whar all the others must match. Nou' the first draw is under the constraint that it must be a club. This probability would work out iike this: It makes sense that this is a smallerprobabiliti, than that of getting 5 cards ofthe same suit. Since we are looking fbr amore selective outcome. we exDect a lo'uverprobability. The first card can be anysuit, so the probability forthis tnal is one. p(5 ctubsr= 11 * 12 '? Il * l0 * 9 =6***)51 51 50 49 46 This next problem is more complicated. and your textbook mav not contain examples like it. If you'refeelins comfortable so far. go ahead and take a look and see if l'ou can fbllow along ...Let's say that we're dealt three cards, and I w'ant to know the probability of eetting at least two cards ofthe same suit.In this situation. we have several things to consider. First. the problem does not state a specific suit, so thefirst card can be any suit. We also have to consider the phrase "at least two" which su_qgests that eithertwo of the cards have matching suits. or all three match. There is also the issue of the order we are dealtthe cards if only two match. We could have our first two cards matching, or the last two matching, or thefirst and the last card matching. Let's break this problem down into smaller pieces.P(at least two out of three cards are of the same suit) = P(two cards have matching suits or all three match) = P(two cards match suit) + P(all three match) These two events are mutually exclusil.e, but they are not simple events. 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endobj 48 0 obj << /Length 49 0 R >> stream tl,-l F l.l rt l.l r I |"'"tdq F"-"r.hl EF"y ",tt IEEry-'"'-fEFqY '"tt I 11lF"y *it I P(two cards match suit) ----+ To figure this out, we must consider the order of the cards. Remember thatthe first card can be any card, but our matching suit pair could be the firstand second card, the second and third card, or the first and third card. So,there are three situations that would satisfy the requirements of thisexpenment. If we use additron ruie and addthese up, we will get the probabilityof exactly two cards matchin,s suit. 234 425.,-' -'L"' P(two czuds match suit) = The second card must match the first.There are 12 cards that match the firstcard's suit. The third card must notmatch. There are three non-matchingsuits of l3 cards each. so 39 cardsthat don't match. This samereasoning applies to the otheroossible outcomes listed. as difficult to compute. Again. remember that thesuit. but the next two have to match whatever that P(all 22 425 three match suit) - = P(two cards match suit) + P(all three match)=# + #=.:!u=@ t 3r-lLE_]F;-"t hT']lmatch 3'd I[t-lIrlF.,-^rcil Fr-ll'lFilhzffilFr-ll'lf"r.hlfl P(ali three cards match suit) --+ This probability isn'tfirst card may be anyfirst cerd "decides." I t2 I | 11 | nXl I Xl | -I st I I so | -TI5 t rnrrch I t -^rch I So, the probabrlity of getting at least two cards of the same suit when three are dealt is...P(at least two out of three cards are of the same suit) = P(two cards have matching suits or all three match) t8 6 of 6 revised 12/19106 endstream endobj 49 0 obj 1507 endobj 50 0 obj << /Length 27 >> stream % CANON_PFINF_TYPE0_TEXTON endstream endobj 51 0 obj << /Length 45 >> stream q 612.24 0 0 791.76 0.00 0.00 cm /Obj45 Do Q endstream endobj 52 0 obj << /Type /Page /MediaBox [ 0 0 612.24 791.76 ] /Parent 11 0 R /Resources << /Font << /F3 5 0 R >> /XObject << /Obj45 45 0 R >> /Subtype 48 0 R /ProcSet [ /PDF /Text /ImageB /ImageC /ImageI ] >> /Contents [ 50 0 R 51 0 R 46 0 R ] >> endobj 11 0 obj << /Kids [ 10 0 R 20 0 R 28 0 R 36 0 R 44 0 R 52 0 R ] /Count 6 /Type /Pages >> endobj 12 0 obj << /Pages 11 0 R /Type /Catalog >> endobj xref 0 53 0000000000 65535 f 0000000015 00000 n 0000000110 00000 n 0000061880 00000 n 0000079929 00000 n 0000079951 00000 n 0000080054 00000 n 0000082437 00000 n 0000082458 00000 n 0000082536 00000 n 0000082631 00000 n 0000451645 00000 n 0000451746 00000 n 0000082880 00000 n 0000138501 00000 n 0000157549 00000 n 0000157572 00000 n 0000160014 00000 n 0000160036 00000 n 0000160115 00000 n 0000160212 00000 n 0000160467 00000 n 0000213404 00000 n 0000230396 00000 n 0000230419 00000 n 0000232610 00000 n 0000232632 00000 n 0000232711 00000 n 0000232808 00000 n 0000233063 00000 n 0000283218 00000 n 0000301789 00000 n 0000301812 00000 n 0000304093 00000 n 0000304115 00000 n 0000304194 00000 n 0000304291 00000 n 0000304546 00000 n 0000368335 00000 n 0000389901 00000 n 0000389924 00000 n 0000392668 00000 n 0000392690 00000 n 0000392769 00000 n 0000392866 00000 n 0000393121 00000 n 0000436202 00000 n 0000449607 00000 n 0000449630 00000 n 0000451192 00000 n 0000451214 00000 n 0000451293 00000 n 0000451390 00000 n trailer << /Size 53 /Info 1 0 R /Root 12 0 R /ID [<1c7cf4a398d258998740249c00c29c4a><1c7cf4a398d258998740249c00c29c4a>] >> startxref 451801 %%EOF