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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.

Here S = {1, 2, 3, 4, 5, 6}.
Let X denote the number obtained on the throw. Then X is a random variable which can take values 1, 2, 3, 4, 5 or 6.
Also, P(1) = P(2) = P(3) = P(4) = P(5) = P(6) = $1 over 6$
∴ the probability distribution of X is

Now, $straight E left parenthesis straight X right parenthesis space equals space sum from straight i space equals space 1 to straight n of space straight x subscript straight i space straight p left parenthesis straight x subscript straight i right parenthesis space equals space 1 space cross times 1 over 6 space plus space 2 space cross times 1 over 6 plus 3 space cross times 1 over 6 plus 4 space cross times 1 over 6 space plus space 5 space cross times space 1 over 6 space plus space 6 space cross times space 1 over 6 space space space space$

$equals space 1 over 6 plus 2 over 6 plus 3 over 6 plus 4 over 6 plus 5 over 6 plus 6 over 6 space equals 21 over 6$

Also, $straight E left parenthesis straight X right parenthesis squared space equals space 1 squared cross times 1 over 6 plus 2 squared cross times 1 over 6 plus 3 squared cross times 1 over 6 plus 4 squared cross times 1 over 6 plus 5 squared cross times 1 over 6 plus 6 squared cross times 1 over 6$

$equals space 1 over 6 plus 4 over 6 plus 9 over 6 plus 16 over 6 plus 25 over 6 plus 36 over 6 space equals 91 over 6$

$therefore space space space space Var left parenthesis straight X right parenthesis space equals space straight E left parenthesis straight X right parenthesis squared space minus space left parenthesis straight E left parenthesis straight X right parenthesis right parenthesis squared$

$equals space 91 over 6 minus open parentheses 21 over 6 close parentheses squared space equals space 91 over 6 minus 441 over 36 space equals space fraction numerator 546 minus 441 over denominator 36 end fraction space equals 105 over 36 space equals 35 over 12.$

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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.

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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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