Question 6 . of 25 Step 1 of 1
Suppose that you and a friend are playing cards and decide to make a bet. If you draw three non-face cards, where a face card is a Jack, a Queen, or a
King, in succession from a standard deck of 52 cards with replacement, you win . Otherwise, you pay your friend . If the same bet was made
30 times, how much would you expect to win or lose? Round your answer to the nearest cent, if necessary.
Answer 2 Points
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Question

Question 6 . of 25 Step 1 of 1
Suppose that you and a friend are playing cards and decide to make a bet. If you draw three non-face cards, where a face card is a Jack, a Queen, or a
King, in succession from a standard deck of 52 cards with replacement, you win

. Otherwise, you pay your friend

. If the same bet was made
30 times, how much would you expect to win or lose? Round your answer to the nearest cent, if necessary.
Answer 2 Points
Prev

UpStudy AI · Accepted Answer

Let $$ p $$ be the probability of drawing a non-face card in one draw. There are 12 face cards (Jack, Queen, King in each suit) out of 52 cards, so there are

$$
52 - 12 = 40 \text{ non-face cards.}
$$

Thus,

$$
p = \frac{40}{52} = \frac{10}{13}.
$$

Since the draws are with replacement, the probability of drawing three non-face cards in succession is

$$
p^3 = \left(\frac{10}{13}\right)^3 = \frac{1000}{2197}.
$$

In each game:  
- If you win (with probability $$ \frac{1000}{2197} $$), you gain \$50.  
- If you lose (with probability $$ 1 - \frac{1000}{2197} = \frac{1197}{2197} $$), you lose \$50.

The expected gain per game, $$ E $$, is

$$
E = 50\left(\frac{1000}{2197}\right) - 50\left(\frac{1197}{2197}\right) = 50\left(\frac{1000-1197}{2197}\right) = -50\left(\frac{197}{2197}\right).
$$

Calculating the expected gain per game numerically,

$$
-50\left(\frac{197}{2197}\right) \approx -50(0.08966) \approx -4.48.
$$

When the bet is made 30 times, the total expected gain (or loss) is

$$
30 \times (-4.48) \approx -134.40.
$$

Rounding to the nearest cent, you would expect to lose approximately $$-\$134.40$$.
You would expect to lose approximately \$134.40 over 30 games.

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