91Ó°ÊÓ

You are playing a game of chance in which four cards are drawn from a standard deck of 52 cards. You guess the suit of each card before it is drawn. The cards are replaced in the deck on each draw. You pay $1 to play. If you guess the right suit every time, you get your money back and $256.

In statistics, the expected value of the random variable is the probability weighted average of all possible values. The formula for finding expected values:

$E\left(x\right)=Ex.P\left(x\right)$

Let X be the amount of money you profit. Therefore, the x takes the values of $-\$1and\$256$.

We know that there are four suits in a standard pack of 52 cards.

The probability of guessing suits is:

The probability of losing is:

$$\begin{gathered} 1-\frac{1}{256}=1-0.0039 \\ =0.9961 \end{gathered}$$

Therefore, the expected profit of playing the game is:

$$\begin{gathered} (256×0.0039)+(-1×0.09961) \\ =0.0023 \end{gathered}$$