Questions: Part 2 of 2 Completed: 8 of 13 My score: 7.31/13 pts (56.25%) A popular casino table game is three-card poker. One aspect of the game is the "pair plus" bet in which a player is paid a dollar amount for any hand of a pair or better, regardless of the hand the dealer has. The table available below shows the profit and probability of various hands of a player playing the 5 pair plus bet. Complete parts (a) and (b) below. Click the icon to view the profit and probability table. (a) What is the expected profit when playing the 5 pair plus bet in three-card poker? The expected profit is -0.1. (Round to two decimal places as needed.) (b) If you play the game for 4 hours with an average of 35 hands per hour, how much would you expect to lose? Expect to lose (Round to two decimal places as needed.)

Part 2 of 2 Completed: 8 of 13 My score: 7.31/13 pts (56.25%) A popular casino table game is three-card poker. One aspect of the game is the "pair plus" bet in which a player is paid a dollar amount for any hand of a pair or better, regardless of the hand the dealer has. The table available below shows the profit and probability of various hands of a player playing the 5 pair plus bet. Complete parts (a) and (b) below. Click the icon to view the profit and probability table. (a) What is the expected profit when playing the 5 pair plus bet in three-card poker? The expected profit is -0.1. (Round to two decimal places as needed.) (b) If you play the game for 4 hours with an average of 35 hands per hour, how much would you expect to lose? Expect to lose (Round to two decimal places as needed.)
Transcript text: Part 2 of 2 Completed: 8 of 13 My score: 7.31/13 pts (56.25\%) A popular casino table game is three-card poker. One aspect of the game is the "pair plus" bet in which a player is paid a dollar amount for any hand of a pair or better, regardless of the hand the dealer has. The table available below shows the profit and probability of various hands of a player playing the $\$ 5$ pair plus bet. Complete parts (a) and (b) below. Click the icon to view the profit and probability table. (a) What is the expected profit when playing the $\$ 5$ pair plus bet in three-card poker? The expected profit is $-\$ 0.1$. (Round to two decimal places as needed.) (b) If you play the game for 4 hours with an average of 35 hands per hour, how much would you expect to lose? Expect to lose $\$$ $\square$ (Round to two decimal places as needed.)
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Solution

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Solution Steps

Step 1: Calculate the Expected Profit

To find the expected profit per hand, we compute the mean of the profit distribution using the formula:

\[ \text{Mean} = \sum (x_i \cdot p_i) \]

where \(x_i\) are the profit values and \(p_i\) are the corresponding probabilities. Substituting the values:

\[ \text{Mean} = -5 \times 0.743 + 0 \times 0.166 + 5 \times 0.075 + 20 \times 0.03 + 25 \times 0.015 + 40 \times 0.001 = -2.32 \]

Thus, the expected profit per hand is:

\[ \text{Expected profit per hand} = -2.32 \]

Step 2: Calculate the Expected Loss Over 4 Hours

Next, we calculate the expected total loss when playing for 4 hours at an average of 35 hands per hour. The total number of hands played is:

\[ \text{Total hands} = 35 \times 4 = 140 \]

The expected total loss is then calculated as:

\[ \text{Total expected loss} = \text{Total hands} \times \text{Expected profit per hand} = 140 \times -2.32 = -324.80 \]

Final Answer

The expected profit per hand is \(\boxed{-2.32}\) and the expected total loss over 4 hours is \(\boxed{-324.80}\).

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