Question

A game involves selecting a card from a regular $52$-card deck and tossing a coin. The coin is a fair coin and is equally likely to land on heads or tails.

• If the card is a face card, and the coin lands on heads, you win $\backslash (6$

• If the card is a face card, and the coin lands on tails, you win $\backslash )2$

• If the card is not a face card, you lose $\$2,$no matter what the coin shows.

a. Find the expected value for this game (expected net gain or loss).

b. Explain what your calculations indicate about your long-term average profits and losses on this game.

c. Should you play this game to win money?

Short answer

(a) The expected loss is $-\frac{8}{13}$.

(b) Calculations indicate that in the long term there will be a loss.

(c) No, I should not play this game.

Step-by-step solution

Given information (part a)

If a card is a face card and the coin lands on the head, then you win $\$6.$

If a card is a face card and the coin lands on tails, then you win $\$2.$

If a card is not a face card, then you lose $\$2.$

Explanation (part a)

$X$	$P\left(x\right)$	$XP\left(x\right)$
$6$	$\frac{12}{52}\times \frac{1}{2}=\frac{3}{26}$	$6\times \frac{3}{26}=\frac{9}{13}$
$2$	$\frac{12}{52}\times \frac{1}{2}=\frac{3}{26}$	$2\times \frac{3}{26}=\frac{3}{13}$
$-2$	$\frac{40}{52}\times 1=\frac{10}{13}$	$-2\times \frac{20}{13}=-\frac{3}{13}$
Total	$1$	$-\frac{8}{13}$

$$\begin{gathered} E\left(x\right)=\underset{}{\sum \left[X\times P\left(x\right)\right]} \\ E\left(x\right)=6\times \frac{9}{13}+2\times \frac{3}{13}-2\times \frac{-20}{13} \\ E\left(x\right)=-\frac{8}{13} \end{gathered}$$

There is an Expected loss of$-8/13$.

Given information (part b)

If a card is a face card and the coin lands on the head, then you win $\$6.$

If a card is a face card and the coin lands on tails, then you win $\$2.$

If a card is not a face card, then you lose$\$2.$

Explanation (part b)

This indicates that in long term there will be a loss because the probability of getting a loss is more than getting a profit. So, there will be a loss.

Given information (part c)

If a card is a face card and the coin lands on the head, then you win$\$6$.

If a card is a face card and the coin lands on tails, then you win $\$2.$

If a card is not a face card, then you lose$\$2.$

Explanation (part c)

NO, I should not play this game as we have already seen in the above part that there is a higher probability of loss. So, we will not play this game.