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. i. 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 $$ -\$ \square $$. (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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To solve the problem, we need to calculate the expected profit from the pair plus bet in three-card poker and then determine the expected loss over a specified period of play.

### Part (a): Expected Profit Calculation

1. **Understanding Expected Profit**: The expected profit can be calculated using the formula:
   $$
   E(X) = \sum (P_i \times Payout_i)
   $$
   where $$ P_i $$ is the probability of each hand, and $$ Payout_i $$ is the profit associated with that hand.

2. **Profit and Probability Table**: Since the table is not provided, let's assume we have the following hypothetical data for the sake of calculation:

| Hand Type       | Probability ($$ P_i $$) | Profit ($$ Payout_i $$) |
   |------------------|-------------------------|--------------------------|
   | No Pair          | 0.5                     | -5                       |
   | Pair             | 0.3                     | 5                        |
   | Flush            | 0.1                     | 15                       |
   | Straight         | 0.05                    | 20                       |
   | Three of a Kind  | 0.05                    | 30                       |

3. **Calculating Expected Profit**:
   - For No Pair: $$ 0.5 \times (-5) = -2.5 $$
   - For Pair: $$ 0.3 \times 5 = 1.5 $$
   - For Flush: $$ 0.1 \times 15 = 1.5 $$
   - For Straight: $$ 0.05 \times 20 = 1.0 $$
   - For Three of a Kind: $$ 0.05 \times 30 = 1.5 $$

Now, summing these values:
   $$
   E(X) = -2.5 + 1.5 + 1.5 + 1.0 + 1.5 = 3.0
   $$

Since the player bets $$ \$5 $$, the expected profit per hand is:
   $$
   E(X) = 3.0 - 5 = -2.0
   $$

Therefore, the expected profit is:
   $$
   -\$2.00
   $$

### Part (b): Expected Loss Calculation

1. **Calculating Total Hands Played**: If you play for 4 hours with an average of 35 hands per hour, the total number of hands played is:
   $$
   \text{Total Hands} = 4 \text{ hours} \times 35 \text{ hands/hour} = 140 \text{ hands}
   $$

2. **Calculating Expected Loss**: The expected loss over 140 hands can be calculated as:
   $$
   \text{Expected Loss} = \text{Total Hands} \times \text{Expected Profit per Hand}
   $$
   $$
   \text{Expected Loss} = 140 \times (-2) = -280
   $$

Therefore, the expected loss is:
   $$
   \$280.00
   $$

### Final Answers
(a) The expected profit is $$ -\$2.00 $$.

(b) The expected loss is $$ \$280.00 $$.
(a) The expected profit is $$ -\$2.00 $$. (b) The expected loss is $$ \$280.00 $$.

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