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Probability

Short Answer Type

961. An urn contains 3 white and 4 red balls. 3 balls are drawn one by one with replacement. Find the probability distribution of the number of red balls.

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962. Find the probability distribution of green balls drawn when 3 balls are drawn one by one without replacement from a bag containing 3 green and 5 white balls.

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963. Find the probability distribution of white balls drawn when 3 balls are drawn one by one without replacement from a bag containing 4 white and 6 red balls.

Let X denote the random variable ‘number of white balls’. X can take the values 0, 1, 2, 3.
Let p be the probability of getting first white ball and q the probability of not getting first white ball.
$therefore space space space space space straight p space space equals space 4 over 10 comma space space space straight q space equals space 1 space minus space straight p space equals space 1 space minus space 4 over 10 space equals space 6 over 10$
$straight P left parenthesis straight X space equals space 0 right parenthesis space equals space straight q space straight q space straight q space space equals space 6 over 10 cross times 5 over 9 cross times 4 over 8 space equals space 5 over 30 straight P left parenthesis straight X space equals space 1 right parenthesis space equals space straight p space straight q space straight q space plus space straight q space straight p space straight q space space plus space straight q space space straight q space straight p space space space space space space space space space space space space space space equals space 4 over 10 cross times 6 over 9 cross times 5 over 8 plus 6 over 10 cross times 4 over 9 cross times 5 over 8 plus 6 over 10 cross times 5 over 9 cross times 4 over 8 space equals space 5 over 30 plus 5 over 30 plus 5 over 30 space equals space 15 over 30 straight P left parenthesis straight X space equals space 2 right parenthesis space equals space straight p space straight p space straight q space plus space straight p space straight q space straight p space space plus space straight q space straight p space space straight p space space space space space space space space space space space space space space space space space equals space 4 over 10 cross times 3 over 9 cross times 6 over 8 plus 4 over 10 cross times 6 over 9 cross times 3 over 8 plus 6 over 10 cross times 4 over 9 cross times 3 over 8 space equals space 3 over 30 plus 3 over 30 plus 3 over 30 space equals 9 over 30 straight P left parenthesis straight X space equals space 3 right parenthesis space equals space straight p space straight p space straight p space equals space 4 over 10 cross times 3 over 9 cross times 2 over 8 space equals space 1 over 30$
∴ probability distribution is

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964. A bag contains 2 white, 3 red and 4 blue balls. Two balls are drawn at random from the bag. If the random variable X denotes the number of white balls among the two balls drawn, describe the probability distribution of X.

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Long Answer Type

965. Two bad eggs are mixed accidentally with 10 good one. Find the probability distribution of the number of bad eggs in 3, drawn at random, from this lot.

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966. From a lot of 30 bulbs which include 6 defectives, a sample of 4 bulbs is drawn at random with replacement. Find the probability distribution of the number of defective bulbs.

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967. Find the probability distribution of number of doublets in three throws of a pair of dice.

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968. A coin is biased so that the head is 3 times as likely to occur as tail. If the coin is tossed twice, find the probability distribution of number of tails.

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969. Three cards are drawn successively with replacement from a well-shuffled deck of 52 cards. A random variable X denotes the number of spades in three cards. Determine the probability distribution of X.

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Short Answer Type

970. Find the probability distribution of the number of successes in two tosses of a die, where a success is defined as:
number greater than 4

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