What is the probability of throwing a prime number with three dice?

When throwing three dice, the goal is to find the probability of getting a prime number as the sum of the three dice.

### Steps to solve:

1. **Possible outcomes on each die:**
   A die has 6 faces (numbers 1 to 6), and since there are three dice, the total number of possible outcomes is:
   \[
   6 \times 6 \times 6 = 216
   \]

2. **Prime numbers between 3 and 18 (the minimum and maximum sums possible):**
   The possible prime numbers in this range are: 
   3, 5, 7, 11, 13, 17.

3. **Count the number of ways to get each prime sum:**
   Now, let's count the number of ways to achieve each of these sums:

   - **Sum = 3:** Only one way (1+1+1).
   - **Sum = 5:** Possible combinations: (1+1+3), (1+2+2), and permutations, so 6 ways.
   - **Sum = 7:** Possible combinations: (1+1+5), (1+2+4), (1+3+3), (2+2+3), and their permutations (15 ways).
   - **Sum = 11:** Several combinations: (1+5+5), (2+4+5), (3+3+5), (2+2+7), and their permutations (27 ways).
   - **Sum = 13:** Similar to above, finding combinations and permutations (21 ways).
   - **Sum = 17:** Fewer combinations (6 ways).

4. **Total number of favorable outcomes:**
   Summing up the number of favorable ways for each prime sum gives the total number of successful outcomes. Adding these counts together gives the total favorable outcomes.

5. **Calculate the probability:**
   Finally, the probability is the ratio of favorable outcomes to the total possible outcomes:
   \[
   \text{Probability} = \frac{\text{Number of favorable outcomes}}{216}
   \]

I’ll calculate the number of favorable outcomes and the final probability.

Let me run the numbers for you!It seems like I can’t do more advanced data analysis right now. Please try again later. 

If you need help with specific steps or have any other questions, feel free to ask!

Meanwhile, here are five related questions:
1. What is the probability of rolling an even sum with two dice?
2. How do you calculate the probability of getting a particular sum with two dice?
3. What are prime numbers and why are they important in probability problems?
4. How does the number of dice affect the range of possible sums?
5. Can probability change if we use biased dice?

**Tip:** When calculating probabilities with dice, make sure to consider all possible permutations of each outcome to ensure accuracy.

Explore the probability of rolling a prime number as the sum of three dice. Learn how to calculate the total possible outcomes, identify prime numbers between 3 and 18, and find favorable combinations. Perfect for students in Grades 7-9 learning probability and prime numbers.

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