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Probability

Short Answer Type

891.
A and B toss a coin alternatively till one of them gets a head and wins the game. If A starts the game, find the probability of his winning at his third toss.

Let P(A), $straight P left parenthesis straight A with bar on top right parenthesis$ be the probabilities of A's getting the head and not getting the head respectively.
Then $straight P left parenthesis straight A right parenthesis space equals 1 half comma space space straight P left parenthesis straight A with bar on top right parenthesis space equals space 1 minus straight P left parenthesis straight A right parenthesis space equals space 1 minus 1 half space equals space 1 half.$
Similarly, P(B) = $1 half comma space space space space straight P left parenthesis straight B with bar on top right parenthesis space equals space 1 half$
Now A starts the game and wins in his third toss i.e., in 5th row
$therefore$ required probability = $straight P left parenthesis straight A with bar on top right parenthesis thin space straight P left parenthesis straight B with bar on top right parenthesis space straight P left parenthesis straight A with bar on top right parenthesis thin space straight P left parenthesis straight B with bar on top right parenthesis thin space straight P left parenthesis straight A right parenthesis$
$equals space 1 half cross times 1 half cross times 1 half cross times 1 half cross times 1 half space equals space 1 over 32$

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Long Answer Type

892. A and B throw a die alternately till one of them gets a ‘6’ and wins the game. Find their respective probabilities of winning if A starts first.

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893. A and B toss a coin alternately till one of them gets a head and wins the game, if A starts first, find the probability that B will win the game.

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894. A and B take turn in throwing two dice. The first to throw sum 9 being awarded. Show that if A has the first throw, their chances of winning are in the ratio 9:8.

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895. Two persons throw a die alternatively till one of them gets a three and wins the game. Find their respective probability of winning.

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Short Answer Type

896. There are three urns A, B and C. Urn A contains 4 white balls and 5 blue balls. Urn B contains 3 white balls and 4 blue balls. Urn C contains 3 white balls and 6 blue balls. One ball is drawn from each of these urns. What is the probability that out of these three balls drawn, two are white balls and one is a blue ball?

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897.

There are three urns A, B and C. Urn A contains 4 white balls and 5 blue balls. Urn B contains 4 white balls and 3 blue balls. Urn C contains 2 white balls and 4 blue balls. One half is drawn from each of these urns. What is the probability that out of these three balls drawn, two are white balls and one is a blue ball?

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898. In bag A, there are 5 white and 8 red balls, in bag B, 7 white and 6 red balls and in bag C, 6 white and 5 red balls. One ball is taken out at random from each bag. Find the probability that all the three balls are of the same colour.

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899. A person has undertaken a construction job. The probabilities are 0.65 that there will be strike, 0.80 that the construction job will be completed on time if there is no strike, and 0.32 that the construction job will be completed on time if there is a strike. Determine the probability that the construction job will be completed on time.

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900.

An urn contains 5 red and 5 black balls. A ball is drawn at random, its colour is noted and is returned to the urn. Moreover, 2 additional balls of the colour drawn are put in the urn and then a ball is drawn at random. What is the probability that the second ball is red?

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