Question

In a gambling game a person draws a single card from an ordinary 52 card playing deck. A person is paid $15 for drawing a jack or a queen and $5 for drawing a king or an ace. A person who draws any other card pays $4. If a person plays this game, what is the expected gain?

Answer 1

There is a \(\frac{8}{52}\) chance of drawing a jack or a queen, and a \(\frac{8}{52}\) chance of drawing a king or an ace. There is a \(\frac{36}{52}\)

chance of drawing any other card.

Let \(X_{\text {be the monetary gain. }}\) \(X=\left\{\begin{array}{c}\$ 15, \frac{8}{52} \\ \$ 5, \frac{8}{52} \\ -\$ 4, \frac{36}{52}\end{array}\right\}\)

\(E[X]=(15)\left(\frac{8}{52}\right)+(5)\left(\frac{8}{52}\right)+(-4)\left(\frac{36}{52}\right)=\$ 0.30\)