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Probability

Question
CBSEENMA12033806

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.

Solution

Let X denote the random variable ‘number of red balls’. X can take the values 0, 1, 2, 3.
Let p be the probability of getting red ball.
           therefore            straight p space equals space 4 over 7 comma space space space straight q space equals space 1 minus space straight p space space equals space 1 minus space 4 over 7 space equals space 3 over 7
straight P left parenthesis straight X space space equals 0 right parenthesis space equals space straight q space straight q space straight q space equals space 3 over 7 cross times 3 over 7 cross times 3 over 7 space equals space 27 over 343
straight P left parenthesis straight X space equals space 1 right parenthesis space equals space space straight p space straight q space straight q space space plus space straight q space straight p space straight q space plus space straight q space straight q space straight p
space space space space space space space space space space space space space space space space space space equals space 4 over 7 cross times 3 over 7 cross times 3 over 7 space plus space 3 over 7 cross times 4 over 7 cross times 3 over 7 plus 3 over 7 cross times 3 over 7 cross times 4 over 7 space equals space 36 over 343 plus 36 over 343 plus 36 over 343 space equals space 108 over 343
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 straight p space
space space space space space space space space space space space space space space space space space equals space 4 over 7 cross times 4 over 7 cross times 3 over 7 plus 4 over 7 cross times 3 over 7 cross times 4 over 7 plus 3 over 7 cross times 4 over 7 cross times 4 over 7 space equals space 48 over 343 plus 48 over 343 plus 48 over 343 space equals space 144 over 343
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 7 cross times 4 over 7 cross times 4 over 7 space equals space 64 over 343
∴ probability distribution is

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