Answer :
To determine the expected return per roll when rolling three fair 6-sided dice and having the given reward/loss conditions, we need to consider all possible outcomes and the corresponding probabilities. Here's a detailed step-by-step solution:
### Step 1: Determine the Total Number of Possible Outcomes
When rolling three 6-sided dice, the total number of possible outcomes is calculated by:
[tex]\[ 6 \times 6 \times 6 = 6^3 = 216 \][/tex]
Thus, there are 216 possible outcomes when rolling three dice.
### Step 2: Calculate Probabilities and Rewards for Each Case
There are three distinct cases to consider based on the roll results and the given conditions:
#### Case 1: All Three Dice Show the Same Number
In this scenario, all three dice show the same number (e.g., 1, 1, 1 or 2, 2, 2, etc.). There are 6 possible ways this can happen (one for each possible number on the dice).
[tex]\[ \text{Probability (all same)} = \frac{6 \text{ outcomes}}{216 \text{ possible outcomes}} = 0.0278 \][/tex]
[tex]\[ \text{Reward} = \$20 \][/tex]
#### Case 2: Exactly Two of the Numbers Are the Same
In this scenario, exactly two of the three dice show the same number and the third die shows a different number (e.g., 1, 1, 2 or 2, 2, 3, etc.).
To calculate the number of outcomes:
- Choose the number that appears twice: 6 ways.
- Choose the die that shows the different number: 5 remaining choices.
- Choose which die shows the different number: 3 ways.
Therefore, the number of favorable outcomes is:
[tex]\[ 3 \text{ (different die positions)} \times 6 \text{ (choices for pair)} \times 5 \text{ (choices for different number)} = 90 \][/tex]
[tex]\[ \text{Probability (two same)} = \frac{90 \text{ outcomes}}{216 \text{ possible outcomes}} = 0.4167 \][/tex]
[tex]\[ \text{Reward} = \$10 \][/tex]
#### Case 3: All of the Numbers Are Different
In this scenario, all three dice show different numbers (e.g., 1, 2, 3 or 2, 4, 5, etc.).
To calculate the number of outcomes:
- Choose three different numbers from six dice: [tex]\( \binom{6}{3} \text{ ways} \)[/tex].
- The number of permutations of these 3 distinct numbers: [tex]\(3! \text{ ways (factorial of 3)} = 6\)[/tex].
Therefore, the number of favorable outcomes is:
[tex]\[ 6 \text{ (choices from 6 numbers)} \times 5 \text{ (remaining 5 numbers)} \times 4 \text{ (remaining 4 numbers)} = 120 \][/tex]
[tex]\[ \text{Probability (all different)} = \frac{120 \text{ outcomes}}{216 \text{ possible outcomes}} = 0.5556 \][/tex]
[tex]\[ \text{Reward} = -\$2 \][/tex]
### Step 3: Calculate the Expected Return
Using the probabilities and rewards for each case, we can calculate the expected return:
[tex]\[ E(\text{return}) = (\text{Probability (all same)} \times \text{Reward (all same)}) + (\text{Probability (two same)} \times \text{Reward (two same)}) + (\text{Probability (all different)} \times \text{Reward (all different)}) \][/tex]
Substituting the values:
[tex]\[ E(\text{return}) = (0.0278 \times 20) + (0.4167 \times 10) + (0.5556 \times (-2)) \][/tex]
[tex]\[ E(\text{return}) = 0.556 + 4.167 - 1.111 = 3.612 \][/tex]
### Step 4: Round to the Nearest Cent
The expected return rounded to the nearest cent is:
[tex]\[ \text{Expected return} = \$3.61 \][/tex]
So, the expected return per roll is [tex]\(\boxed{3.61}\)[/tex] dollars.
### Step 1: Determine the Total Number of Possible Outcomes
When rolling three 6-sided dice, the total number of possible outcomes is calculated by:
[tex]\[ 6 \times 6 \times 6 = 6^3 = 216 \][/tex]
Thus, there are 216 possible outcomes when rolling three dice.
### Step 2: Calculate Probabilities and Rewards for Each Case
There are three distinct cases to consider based on the roll results and the given conditions:
#### Case 1: All Three Dice Show the Same Number
In this scenario, all three dice show the same number (e.g., 1, 1, 1 or 2, 2, 2, etc.). There are 6 possible ways this can happen (one for each possible number on the dice).
[tex]\[ \text{Probability (all same)} = \frac{6 \text{ outcomes}}{216 \text{ possible outcomes}} = 0.0278 \][/tex]
[tex]\[ \text{Reward} = \$20 \][/tex]
#### Case 2: Exactly Two of the Numbers Are the Same
In this scenario, exactly two of the three dice show the same number and the third die shows a different number (e.g., 1, 1, 2 or 2, 2, 3, etc.).
To calculate the number of outcomes:
- Choose the number that appears twice: 6 ways.
- Choose the die that shows the different number: 5 remaining choices.
- Choose which die shows the different number: 3 ways.
Therefore, the number of favorable outcomes is:
[tex]\[ 3 \text{ (different die positions)} \times 6 \text{ (choices for pair)} \times 5 \text{ (choices for different number)} = 90 \][/tex]
[tex]\[ \text{Probability (two same)} = \frac{90 \text{ outcomes}}{216 \text{ possible outcomes}} = 0.4167 \][/tex]
[tex]\[ \text{Reward} = \$10 \][/tex]
#### Case 3: All of the Numbers Are Different
In this scenario, all three dice show different numbers (e.g., 1, 2, 3 or 2, 4, 5, etc.).
To calculate the number of outcomes:
- Choose three different numbers from six dice: [tex]\( \binom{6}{3} \text{ ways} \)[/tex].
- The number of permutations of these 3 distinct numbers: [tex]\(3! \text{ ways (factorial of 3)} = 6\)[/tex].
Therefore, the number of favorable outcomes is:
[tex]\[ 6 \text{ (choices from 6 numbers)} \times 5 \text{ (remaining 5 numbers)} \times 4 \text{ (remaining 4 numbers)} = 120 \][/tex]
[tex]\[ \text{Probability (all different)} = \frac{120 \text{ outcomes}}{216 \text{ possible outcomes}} = 0.5556 \][/tex]
[tex]\[ \text{Reward} = -\$2 \][/tex]
### Step 3: Calculate the Expected Return
Using the probabilities and rewards for each case, we can calculate the expected return:
[tex]\[ E(\text{return}) = (\text{Probability (all same)} \times \text{Reward (all same)}) + (\text{Probability (two same)} \times \text{Reward (two same)}) + (\text{Probability (all different)} \times \text{Reward (all different)}) \][/tex]
Substituting the values:
[tex]\[ E(\text{return}) = (0.0278 \times 20) + (0.4167 \times 10) + (0.5556 \times (-2)) \][/tex]
[tex]\[ E(\text{return}) = 0.556 + 4.167 - 1.111 = 3.612 \][/tex]
### Step 4: Round to the Nearest Cent
The expected return rounded to the nearest cent is:
[tex]\[ \text{Expected return} = \$3.61 \][/tex]
So, the expected return per roll is [tex]\(\boxed{3.61}\)[/tex] dollars.
