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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.

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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.

Let p be the probability of obtaining a head when a coin is tossed once and q that of obtaining a tail
From given condition,
p = 3 q, p + q = 1
$therefore space space space space space 3 space straight q space plus space straight q space equals space 1 space space space space space space space space space space space space space space space rightwards double arrow space space space space 4 space straight q space equals space 1 space space space space space rightwards double arrow space space space space straight q space equals space 1 fourth therefore space space space space straight p space equals space 3 space straight q space equals space 3 over 4.$
Let X denote the number of tails in two tosses of the coin, then X can take values 0, 1 and 2.
$straight P left parenthesis straight X space equals 0 right parenthesis space equals space space straight P left parenthesis no space tail right parenthesis space equals space straight P left parenthesis HH right parenthesis space equals space straight p space straight p space space equals space 3 over 4 cross times 3 over 4 space equals 9 over 16 straight P left parenthesis straight X space equals space 1 right parenthesis space equals space straight P left parenthesis one space tail space and space one space head right parenthesis space space space space space space space space space space space space space space space space space equals space straight P left parenthesis HT comma space TH right parenthesis space equals space straight P left parenthesis HT right parenthesis space plus space straight P left parenthesis TH right parenthesis space space space space space space space space space space space space space space space space space equals space straight p space straight q space plus space straight q space straight p space space equals space 2 space straight p space straight q space equals space 2 cross times 3 over 4 cross times 1 fourth space equals 6 over 16$

and $straight P left parenthesis straight X space equals space 2 right parenthesis space equals space straight P left parenthesis two space tails right parenthesis space equals space straight P left parenthesis TT right parenthesis space equals space qq space equals 1 fourth cross times 1 fourth space equals 1 over 16$
∴ probability distribution of X is

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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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