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Question
In a dice game, a player pays a stake of Rs 1 for each throw of a die. She receives Rs 5 if the die shows a 3, Rs 2 if the die shows a 1 or 6, and nothing otherwise. What is the player’s expected profit per throw over a long series of throws?
Solution
Let X be the random variable of profit per throw.
| X | – 1 | 1 | 4 |
| P(X) | `1/2` | `1/3` | `1/6` |
Since, she loses ₹ 1 for getting any of 2, 4, 5.
So, P(X = – 1) = `1/6 + 1/6 + 1/6`
= `3/6`
= `1/2`
P(X = 1) = `1/6 + 1/6`
= `2/6`
= `1/3` .....(∵ Die showing 1 or 6)
P(X = 4) = `1/6` .....(∵ Die shows only a 3)
Player’s expected profit = `sum"p"_1x"i"`
= `-1 xx 1/2 + 1 xx 1/3 + 4 xx 1/6`
= `-1/2 + 1/3 + 2/3`
= `1/2`
= ₹ 0.50
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