Suppose two players, A and B, are playing a sport (like ping-pong for example) w

Question

Suppose two players, A and B, are playing a sport (like ping-pong for example) where the winner is determined by playing a series of “games”, and where the winner is determined as “the first person to win 3 games”. Let’s assume A and B are of the same strength, so that the probability of A [and also B] winning an individual game is .5. Answer the following questions:

a)what is the probability that the A wins the series in 3 games

b) what is the probability that A wins the series in 4 games

c) what is the probability that A wins in 5 games

d) what is the probability that the series will “go to five games” in order to determine the winner?

e) as d), for “4 games” and “3 games”

f) to check the correctness of your work, you should verify that the probabilities of “the series goes to 5 games” , “....4 games”, and “.... 3 games” add up to 1

Explanation / Answer

a) for this A should win all 3 games, so probability of A winning all 3 games is 0.5*0.5*0.5 = 0.125

b)for this A has to win any 2 games out of 3 and the 4th game= choosing 2 out of first 3 games on which A won then A winning two game and losing one out of first 3 games and winning 4th game= 3*(0.5^2)*(0.5)*(0.5)=0.1875

c)for this A has to win any 2 games out of 4 and the 5th game=choosing 2 out of first 4 games on which A won A winning two game and losing two out of first 4 games and winning 5th game= 6*(0.5^2)*(0.5^2)*(0.5)=0.01875

d)for series to go to 5 games =P(A winning the series in 5 game)+P(B winning the series in 5 games)

also P(A winning the series in 5 game) = P(B winning the series in 5 games)

so or series to go to 5 games = 2* P(A winning the series in 5 game)=2*0.1875(from part c)=0.375

e)for series to go to 4 games =P(A winning the series in 4 game)+P(B winning the series in 4 games)

also P(A winning the series in 4 games) = P(B winning the series in 4 games)

so or series to go to 4 games = 2* P(A winning the series in 4 game)=2*0.1875(from part b)=0.375

e)for series to go to 3 games =P(A winning the series in 3 game)+P(B winning the series in 3 games)

also P(A winning the series in 3 game) = P(B winning the series in 3 games)

so or series to go to 3 games = 2* P(A winning the series in 3 game)=2*0.125(from part c)=0.25

f)0.375+0.375+0.25=1

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