Question:
In a dice game, a player pays a stake of Re1 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’s take X to be the random variable of profit per throw.
As, she loses Rs 1 for giving any od 2, 4, 5.
So, P(X = -1) = 1/6 + 1/6 + 1/6 = 3/6 = ½
P(X = 1) = 1/6 + 1/6 = 2/6 = 1/3 [Since, die showing 1 or 6]
P(X = 4) = 1/6 [Since, die shows only a 3]
Thus, the player’s expected profit = Ʃp1xi
= -1 x ½ + 1 x 1/3 + 4 x 1/6 = -1/2 + 1/3 + 2/3 = ½ = Rs 0.50