Question

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.

Solution

Let X denote the random variable ‘number of green balls’. X can take the values 0, 1, 2, 3.
Let p be the probability of getting first green ball and q the probability of not getting first green ball.
$therefore space space space straight p space equals space 3 over 8 comma space space space space straight q space equals space 1 minus straight p space equals 1 minus 3 over 8 space equals 5 over 8$
$straight P left parenthesis straight X space equals 0 right parenthesis space equals space straight q space straight q space straight q space space equals 5 over 8 cross times 4 over 7 cross times 3 over 6 space equals space 10 over 56$
$straight P left parenthesis straight X space equals space 1 right parenthesis space equals space straight p space straight q space straight q space space plus space straight q space straight p space straight q space plus space straight q space straight q space straight p space space space space space space space space space space space space space space space space space space equals space 3 over 8 cross times 5 over 7 cross times 4 over 6 plus 5 over 8 cross times 3 over 7 cross times 4 over 6 plus 5 over 8 cross times 4 over 7 cross times 3 over 6 space equals 10 over 56 plus 10 over 56 plus 10 over 56 equals 30 over 56 straight P left parenthesis straight X space equals space 2 right parenthesis space equals space straight p space straight p space straight q space plus space straight p space straight q space straight p space plus space straight q space straight p space straight p space space space space space space space space space space space space space space space space space space space equals space 3 over 8 cross times 2 over 7 cross times 5 over 6 plus 3 over 8 cross times 5 over 7 cross times 2 over 6 plus 5 over 8 cross times 3 over 7 cross times 2 over 6 space equals 5 over 56 plus 5 over 56 plus 5 over 56 space equals 15 over 56 straight P left parenthesis straight X space equals space 3 right parenthesis space equals space straight p space straight p space straight p space space equals space 3 over 8 cross times 2 over 7 cross times 1 over 6 space equals space 1 over 56$
∴ probability distribution is

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Probability

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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