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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.

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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.

Let S denote the success (getting a '6') and F denote the failure (not getting a '6').
$therefore space space space straight P left parenthesis straight S right parenthesis space equals 1 over 6 comma space space straight P left parenthesis straight F right parenthesis space equals space 5 over 6$
P(A wins in the first throw) = P(S)= $1 over 6$
A gets the throw, when the first throw by A and second throw by B result into failures. Therefore,
$straight P left parenthesis straight A space wins space in space the space third space throw right parenthesis space equals space straight P left parenthesis FFS right parenthesis space equals 5 over 6.5 over 6.1 over 6 space equals open parentheses 5 over 6 close parentheses squared space open parentheses 1 over 6 close parentheses straight P left parenthesis straight A space wins space in space the space 5 th space throw right parenthesis space equals space straight P left parenthesis FFFFS right parenthesis space equals space open parentheses 5 over 6 close parentheses to the power of 4 space open parentheses 1 over 6 close parentheses space and space so space on. space$
$Hence space straight P left parenthesis straight A space wins right parenthesis space equals space 1 over 6 space plus open parentheses 5 over 6 close parentheses squared space open parentheses 1 over 6 close parentheses space plus space open parentheses 5 over 6 close parentheses to the power of 4 space open parentheses 1 over 6 close parentheses plus.... space equals space fraction numerator begin display style 1 over 6 end style over denominator 1 minus begin display style 25 over 36 end style end fraction space space space space space open square brackets because space space straight S space equals space fraction numerator straight a over denominator 1 minus straight r end fraction close square brackets$
$equals space 6 over 11$
P(B wins) = 1 - P(A wins) = $1 minus 6 over 11 space equals 5 over 11.$

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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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