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?

Answer 1

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