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Probability

Short Answer Type

981. In a meeting, 70% of the members favour and 30% oppose a certain proposal. A member is selected at random and we take X = 0 if he opposed, and X = 1 if he is in favour. Find E(X) and Var(X).

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Long Answer Type

982. Find the variance of the number obtained on a throw of an unbiased die.

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983. A die is tossed twice. Getting a number greater than 4 is considered a success. Find the variance of the probability distribution of the number of successes.

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984. Two bad eggs are mixed accidentally with 10 good ones. Three eggs are drawn at random without replacement from this lot. Find the mean and variance for the number of bad eggs.

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985. Three bad eggs got mixed up with 7 good eggs. If three eggs are drawn (without replacement) from 10 eggs, find the mean and variance for the number of bad eggs among them.

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986. Five defective bulbs are accidentally mixed with twenty good ones. It is not possible to just look at a bulb and tell whether or not a bulb is defective. Four bulbs are drawn at random from this lot. Find the mean number of defective bulbs drawn.

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987. Two cards are drawn simultaneously (or successively, without replacement) from a well-shuffled deck of 52 cards. Compute σ² for the number of aces.

Let X denote the number of aces obtained in two draws. X takes the values 0, 1, 2.
Let p denote probability of first ace.
$therefore space space space straight p space equals space 4 over 52 comma space space straight q space equals space 48 over 52$

$straight P left parenthesis straight X space equals space 0 right parenthesis space equals space 48 over 52 cross times 47 over 51 space equals 12 over 13 cross times 47 over 51 space equals space 188 over 221 straight P left parenthesis straight X space equals space 1 right parenthesis space equals space 4 over 52 cross times 48 over 51 plus 48 over 52 cross times 4 over 51 space equals 16 over 221 plus 16 over 221 space equals space 32 over 221 straight P left parenthesis straight X space equals space 2 right parenthesis space equals space 4 over 52 cross times 3 over 51 space equals space 1 over 31 cross times 1 over 17 space equals 1 over 221$
∴ probability distribution is

$therefore space space space space straight X space takes space the space values space 0 comma space 1 comma space 2 space with space probabilities space 188 over 221 comma space 32 over 221 comma space 1 over 221$

$straight mu space equals sum from blank to blank of straight x space straight p subscript straight i space equals space 34 over 221 straight sigma squared space equals sum from blank to blank of straight x squared space straight p space subscript straight i space minus space straight mu squared space equals space 36 over 221 space minus space open parentheses 34 over 221 close parentheses squared space equals space 36 over 221 minus open parentheses 2 over 13 close parentheses squared space equals 36 over 221 minus 4 over 169 space space space space space equals space fraction numerator 6084 minus 884 over denominator 37349 end fraction space equals 5200 over 37349 space equals 400 over 2873$

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988. In a game, a man wins a rupee for a six and loses a rupee for any other number when a fair die is thrown. The man decided to throw a die thrice but to quit as and when he gets a six. Find the expected value of the amount he wins/loses.

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