To decide who gets to eat the last piece of cake, Don and Diane decide to roll two dice. If the sum is 12 or 10, then Don gets the last piece. If the sum of the dice is 2 or 7, then Diane can have the last piece. find the proabbilites of them getting the last piece

To find the probabilities that Don and Diane get the last piece of cake, we need to determine the probabilities of rolling the sums 12 or 10 for Don and 2 or 7 for Diane using two dice. 

First, let's list the possible outcomes when rolling two dice. Each die has 6 faces, so there are a total of \( 6 \times 6 = 36 \) possible outcomes.

### Calculating Don's Probability
Don wins if the sum of the dice is 12 or 10.

- **Sum = 12:** The only combination is (6, 6). 
- **Sum = 10:** The possible combinations are (4, 6), (5, 5), and (6, 4).

Total combinations for Don: \( 1 + 3 = 4 \)

### Calculating Diane's Probability
Diane wins if the sum of the dice is 2 or 7.

- **Sum = 2:** The only combination is (1, 1).
- **Sum = 7:** The possible combinations are (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), and (6, 1).

Total combinations for Diane: \( 1 + 6 = 7 \)

### Calculating Probabilities
The probability of an event is the number of favorable outcomes divided by the total number of possible outcomes.

- **Probability that Don gets the last piece:**
  \[
  P(\text{Don}) = \frac{\text{Number of favorable outcomes for Don}}{\text{Total possible outcomes}} = \frac{4}{36} = \frac{1}{9} \approx 0.111
  \]

- **Probability that Diane gets the last piece:**
  \[
  P(\text{Diane}) = \frac{\text{Number of favorable outcomes for Diane}}{\text{Total possible outcomes}} = \frac{7}{36} \approx 0.194
  \]

Thus, the probabilities are:
- Don: \(\frac{1}{9} \approx 0.111\) or 11.1%
- Diane: \(\frac{7}{36} \approx 0.194\) or 19.4%

Would you like more details on the calculations or have any questions?

Here are 5 questions you might ask:
1. What is the probability of rolling a sum of 7 with two dice?
2. How do you calculate the probability of an event with multiple favorable outcomes?
3. What are the odds in favor of Diane winning?
4. How would the probabilities change if the rules for winning were different?
5. Can you explain the probability distribution for rolling two dice?

**Tip:** When solving probability problems involving dice, it's helpful to write out all possible outcomes to ensure that you count the favorable outcomes correctly.

Calculate the probabilities for Don and Diane to win the last piece of cake based on dice rolls. Includes detailed calculations for sums of 12, 10, 2, and 7 with two dice.

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