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, then what is the player's expected profit per throw over a long series of throws?

Let X is the random variable of profit per throw.

Since, she loss Rs.1 on getting any of 2,4 or 5.

So, at $X = - 1,$ $P(X) = \frac{1}{6} + \frac{1}{6} + \frac{1}{6} = \frac{3}{6} = \frac{1}{2}$

Similarly, at $X = 1,$ $P(X) = \frac{1}{6} + \frac{1}{6} = \frac{1}{3}$

[if die shows of either 1 or 6]
and at $X = 4,$ $P(X) = \frac{1}{6}$

[if die shows a 3]
$\therefore$ Player's expected profit $= E(X) = \Sigma XP(X)$

$= - 1 \times \frac{1}{2} + 1 \times \frac{1}{3} + 4 \times \frac{1}{6}$

$= \frac{{ - 3 + 2 + 4}}{6} = \frac{3}{6} = \frac{1}{2} = {\rm{Rs}}.0.50$