Question

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. What is your expected profit of playing the game over the long term?

Short answer

The excepted profit of playing the game over the long term is$0.0023$

Step-by-step solution

Content Introduction

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.

Content Explanation

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:

$$\begin{gathered} \frac{1}{4}\times \frac{1}{4}\times \frac{1}{4}\times \frac{1}{4}=\frac{1}{256} \\ =0.0039 \end{gathered}$$

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:

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