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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.

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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.

Let E be the event of getting 3 in a throw of die.

therefore space space straight P left parenthesis straight E right parenthesis space equals space 1 over 6 comma space space straight P left parenthesis straight E with bar on top right parenthesis space equals space 1 minus straight P left parenthesis straight E right parenthesis space equals space 1 minus 1 over 6 space equals 5 over 6

Any person who starts the game can win in the first throw, 3rd throw, 5th throw and so on.
$therefore$ probability of winning A
$space equals space straight P left parenthesis straight E right parenthesis space plus space straight P left parenthesis straight E with bar on top space straight E with bar on top space straight E right parenthesis space plus space straight P left parenthesis straight E with bar on top space straight E with bar on top space straight E with bar on top space straight E with bar on top straight E right parenthesis plus... equals space 1 over 6 cross times 1 over 6 cross times 5 over 6 cross times 5 over 6 plus 1 over 6 cross times 5 over 6 cross times 5 over 6 cross times 5 over 6 cross times 5 over 6 plus... equals space 1 over 6 open square brackets 1 plus open parentheses 5 over 6 close parentheses squared plus open parentheses 5 over 6 close parentheses to the power of 4 plus.... close square brackets equals space 1 over 6 open square brackets fraction numerator 1 over denominator 1 minus begin display style 25 over 36 end style end fraction close square brackets space equals space 1 over 6 cross times 36 over 11 space equals space 6 over 11$
Since either of two person wins
$therefore$ probability of winning B = $1 minus 6 over 11 space equals 5 over 11$

$therefore$ required probabilities are $6 over 11 comma space 5 over 11.$

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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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