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

Solution

 A person who draws any other card pays $4. If a person plays this game, what is the expected gain?

Let y be the money which paid or pays :

Then      \( \begin{cases} Y=15 \hspace{2mm}for \hspace{2mm} Jack \hspace{2mm} or \hspace{2mm} queen & \quad \\ Y=5 \hspace{2mm}for\hspace{2mm} king\hspace{2mm} or\hspace{2mm} ace & \quad \\ Y=-4 \hspace{2mm} otherwise& \quad \end{cases} \)

\( \implies P(jack\hspace{2mm} or\hspace{2mm} queen)=\frac{8}{52}\hspace{2mm} \)

          \( P(king\hspace{2mm}or\hspace{2mm}ace)=\frac{8}{52}\hspace{2mm} \)

         \( P(otherwise)=\frac{36}{52} \)

Then \( E(Y)=15\times\frac{8}{52}+5\times\frac{8}{52}-4\times\frac{36}{52}=\frac{16}{52}=0.3\ \)

 

Therefore. \( E(Y)=0.3 \)