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Question
Four balls are to be drawn without replacement from a box containing 8 red and 4 white balls. If X denotes the number of red ball drawn, find the probability distribution of X.
Solution
Since 4 balls have to be drawn, therefore, X can take the values 0, 1, 2, 3, 4.
P(X = 0) = P(no red ball) = P(4 white balls)
= `(""^4"C"_4)/(""^12"C"_4) = 1/495`
P(X = 1) = P(1 red ball and 3 white balls)
= `(""^8"C"_1 xx ""^4"C"_3)/(""^12"C"_4) = 32/495`
P(X = 2) = P(2 red balls and 2 white balls)
= `(""^8"C"_2 xx ""^4"C"_2)/(""^12"C"_4) = 168/495`
P(X = 3) = P(3 red balls and 1 white ball)
= `(""^8"C"_3 xx ""^4"C"_1)/(""^12"C"_4) = 224/495`
P(X = 4) = P(4 red balls)
= `(""^8"C"_4)/(""^12"C"_4) = 70/495`
Thus the following is the required probability distribution of X
| X | 0 | 1 | 2 | 3 | 4 |
| P(X) | `1/495` | `32/495` | `168/195` | `224/495` | `70/495` |
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