John and Kyle play a game, which has 3 rounds, and to win, a person must win at least 2⁄3 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??