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QUESTION

Homer and Marge play a game by taking turns rolling a standard six-sided die, starting with Homer, until there occurs a sequence of k or more 4s immediately followed by a 5 or 6 (e.g. 45, 245 or 5446,

Homer and Marge play a game by taking turns rolling a standard six-sided die, starting with Homer, until there occurs a sequence of k or more 4s immediately followed by a 5 or 6 (e.g. 45, 245 or 5446,

Homer and Marge play a game by taking turns rolling a standard six-sided die, starting with Homer, until there occurs a sequence of k or more 4s immediately followed by a 5 or 6 (e.g. 45, 245 or 5446, if k = 1). The last person to roll wins the game. (E.g., if k = 2, then each of the sequences 446, 64445 and 465443445 results in Homer winning.) Find the probability that Homer wins, for each k = 0, 1, 2, 3 and 99.

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***** **** **** ** *** sequence ****** and there *** *** number of rolls ** **** **** ***** ***** be *** **** *** ** roll(either * * ** * ** thus let * ****** *** *** ****** **** *** sequence **** n+1 tosses ***** n ** even the ******** **** **** like:- *********** ******** ** 6) Thus *** ****** *** occupied ***** *** **** ** * ** 6 *** starting *** spaces *** be ******** Thus *** *********** ** Favourable ************** ********* 6^(n-k)*1*2/6^(n+1)=2/6^(k+1) *** *** k **** ****** ************************** ** **

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