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Homework answers / question archive / John and Kyle play a game, which has 3 rounds, and to win, a person must win at least 2⁄3 rounds
John and Kyle play a game, which has 3 rounds, and to win, a person must win at least 2⁄3 rounds. Assume that each round is independent of all other rounds. If Kyle is 3 times more likely to win a round than John, what is...
(a) the probability that John wins the game?
(b) the probability that Kyle wins the game given that John wins the first round?
(c) the probability that John wins the game given that Kyle wins exactly one of the first
two rounds?
John: p_____________ John's chance to win a round.
Kyle: 3p____________ Kyle's chance to win a round.
?→? ?p+3p=1? ?→? ?4p=1? ?→? ?P=1/4?? ?∧? ?3p=3/4??
John wins a round with probability 1/4 and Kyle wins a round with probability 3/4.
each round is independent of the other.
a) For John to win the game, he must win 2 rounds, which occurs in 3 different ways, or win all 3 rounds, which occurs in only one way.
?→? ?P=? ?3.(1/4.1/4.3/4?)+? ?¼.1/4.1/4?? = ?9/64?+1/64?? = ?10/64 =5/?32?? John's chance of winning the game.
b) Probability of kyle winning given that john won the first game.
?→? ?P=? ?¼.3/4.3/4?? = ?9/64
c) For Jhon to win since Kyle won one of the first 2 games, this can happen in 2 ways.
?→? ?P=? ?2(3/4.1/4.1/4?)? = ?2/64 =1/32??
