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Question
A box contains 30 fruits, out of which 10 are rotten. Two fruits are selected at random one by one without replacement from the box. Find the probability distribution of the number of unspoiled fruits. Also find the mean of the probability distribution.
Solution
Total fruits = 30
Out of which 10 are rotten fruits.
∴ 30 – 10 = 20 are unspoiled fruits
Now two fruits are selected at random one by one without replacement.
and Random variable X = Number of unspoiled fruits
= {0, 1, 2}
∴ P(X = 0) = Probability of getting both rotten fruits
= `10/30 xx 9/29`
= `9/87`
P(X = 1) = Probability of getting 1 unspoiled fruits
= `(10/30 xx 20/29) + (20/30 xx 10/29)`
= `40/87`
and P(X = 2) = Probability of getting 2 unspoiled fruits
= `20/30 xx 19/29`
= `38/87`
Hence probability distribution is
| X | 0 | 1 | 2 |
| P(X) | `9/87` | `40/87` | `38/87` |
and Mean = `sumX.P(X)`
= `0 xx 9/87 + 1 xx 40/87 + 2 xx 38/87`
= `0 + 40/87 + 76/87`
= `116/87`
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