Coin Toss - Coin Tossing Games 1) GAME OF TOSSING ONE COIN....

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Coin Tossing Games 1) GAME OF TOSSING ONE COIN. Keep tossing a coin until A or B wins. A wins on contiguous HH, and B wins on HT. Solution. [ ] 1/2, [ ] 1/2 P Awins P B wins = = 2) GAME OF TOSSING ONE COIN. Keep tossing a coin until A or B wins. A wins on contiguous HH, and B wins on TH Solution. [ ] 1/4 , [ ] 3/4 P Awins P B wins = = . 3) GAME OF TOSSING OWN COIN. Each player has their own coin. A wins on HH and B wins on HT (or TH) Solution. [ ] 0.32, [ ] 0.54, [ ] 0.14 P Awins P B win P Aand Btie = = =
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SOLUTION OF GAME (3) 11 ( ): prob that hits for the first time at the -th toss, 1,2, hits for the first time at the -th toss () ( 1 ) A B kk AA BB jj AAA Notations P k A HH k k P k B HT k Q k P j and Q k P j Then P k Qk = = = = −− ∑∑  [ ] [ ] { } { } [ ] { } [ ] ( ) hits at least once at or before the -th toss ) ( 2 ) ()1 () () A AB B A B B A Q k P A HH k P A w i n s Pk Q k P B wins P k Q k P tie P k P k ∞∞ = = = = = = ++ + + = = −=
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Suppose has never hit until the -th toss. ( ): the number of such sequences that end with a . ( ): the number of such sequences that end with a .
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Coin Toss - Coin Tossing Games 1) GAME OF TOSSING ONE COIN....

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