The Minnesota Twins are to play a series of 5 games in Boston against the Red So

Question

The Minnesota Twins are to play a series of 5 games in Boston against the Red Sox. For any one game it is estimated that the probability of a Twins' win is 0.5. The outcomes of the 5 games are independent of one another. (a) What is the probability that the Twins will win all 5 games? (b) What is the probability that the Twins will win a majority of the 5 games? (c) If the Twins win the first game, what is the probability that they will win a majority of the five games? (d) Before the series begin, what is the expected number of Twins' wins in these 5 games? (e) If the Twins win the first game, what is the expected number of Twins' wins in the five-game series?

Explanation / Answer

Let x be the number of wins

p=q=0.5

1) P(x=5) = 0.5^5 = 0.3125

2) P(Majority) = P(3)+P(4)+P(5) = 5C3 * 0.5^3*0.5^2 + 5C4 *0.5^4*0.5^1 + 5c5 * 0.5^5 = (10+5+1)*0.5^5 =16 *0.5^5 = 0.5

3) P( Majority/First) = P( WIn 2 of 4 games ) + P(WIn 3) + P(WIn 4) = 4C2 * 0.5^4 + 4C3 *0.5^4 + 4C4 * 0.5^4 = 11* 0.5^4 = 0.6875

4) E(x) = np = 5*0.5=2.5

5) First game -twins win

Out of 4 games, since games are independent, twins are expected to win 4*0.5=2 games

Total number of games expected to win = 2+1=3

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