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Question
Two dice are thrown simultaneously. If X denotes the number of sixes, find the expectation of X.
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Solution
Here, X represents the number of sixes obtained when two dice are thrown simultaneously. Therefore, X can take the value of 0, 1, or 2.
∴ P (X = 0) = P (not getting six on any of the dice)
= `(5 xx 5)/(6 xx 6)`
= `25/36`
P (X = 1) = P (six on first die and no six on second die) + P (no six on first die and six on second die)
= `2(1/6xx5/6)`
= `10/36`
P (X = 2) = P (six on both the dice) =`1/36`
∴ The required probability distribution is as follows.
| X | 0 | 1 | 2 |
| P(X) | `25/36` | `10/36` | `1/36` |
Expectation of X = E(X) = `sum X_iP(X_i)`
= `0 xx 25/36 + 1 xx10/36 + 2xx 1/36`
= `1/3`
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