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Two athletic teams play a series of games; the first team to win 4 games is declared the overall winner .Suppose that one of the team is stronger than the other and wins each game with prob 0.6,independently of the outcomes of other games.Find the prob for i=4,5,6,7,,,that the sronger team wins the series in exactly i games.Compare the prob that the stronger team wins with the prob that it would win a 2-out-3 series.

+1 vote

Let us do for one value of 'i' and same can be done for other values of 'i' as well..Let us take i = 6 here..

So it means that the 6th game must result to a win in order to win the series..Hence we need to ensure 3 wins in preceding 5 matches..Hence now this problem reduces to binomial distribution B(5 , 0.6) as we are considering the preceding 5 matches(trials) and probability of individual trial = 0.6..

Hence

P(series is won by strong team)   =   P(3 wins in 5 matches) * P(win in 6th match)

=  B(5,0.6)  *  0.6

=  5C3 (0.6)3 (0.4)2 * 0.6

=  0.20736

=  0.21 (correct to 2 decimal places)

Similarly we can find for i = 4 , 5  and 7 as well.

P(win a 2 - out of 3 series)          =  3C2 (0.6)2 (0.4)

=  0.432

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I have a little doubt, why u are not considering the case when one team wins at least 4 games in any order in 6 game series then team wins the series?
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This is an obvious doubt which may come to one's mind in this problem ; the problem with selecting any 4 wins out of 6 is that in that case it is possible that the first four games are won itself ..In that case the team should have won then and there itself as the rule mentioned in the question :

the first team to win 4 games is declared the overall winner

Hence the team wins in the 4th game itself in this scenario..Whereas we want that the team wins after playing exactly 6 games..Thus we have to ensure that the 4th win comes in the 6th game and not before that..Hope this clears your doubt..:)

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okay..I got it.Thanks :)

+1

Ans given in instructer manual is P{wins in i games} = C(i-1 ,3) *(.6)4 *(.4)i-4

Habibkhan your approach is really nice.

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Ya on simplification it will come out to be the same..:)

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+1 vote
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