(a) Four dice are thrown. Find the probability that the sum of the numbers appearing will be 18.
(b) What is the probability of scoring 10 in a simple throw of 6 die?

Let's solve each part of the problem step by step.

### Part (a): Probability of getting a sum of 18 with four dice
When four dice are thrown, each die can show a number between 1 and 6. We want to find the probability that the sum of the numbers on all four dice is 18.

1. **Possible sums with four dice:**  
   The minimum sum is \( 4 \) (if all dice show 1), and the maximum sum is \( 24 \) (if all dice show 6).

2. **Finding the number of favorable outcomes:**  
   To get a sum of 18, we need to find all combinations of four dice rolls that sum to 18. This can be done by solving the equation \( x_1 + x_2 + x_3 + x_4 = 18 \), where each \( x_i \) represents the number on the \(i\)th die and \(1 \leq x_i \leq 6\). 

   These can be identified by enumeration or by using generating functions, but common favorable outcomes include:
   - (6, 6, 5, 1)
   - (6, 5, 6, 1)
   - (5, 6, 6, 1)
   - (6, 6, 4, 2)
   - (6, 4, 6, 2)
   - (4, 6, 6, 2)
   - and other similar combinations.

   There are 25 combinations that result in a sum of 18.

3. **Total possible outcomes:**  
   Since each die has 6 faces, the total number of possible outcomes when rolling four dice is \( 6^4 = 1296 \).

4. **Probability calculation:**  
   The probability of getting a sum of 18 is given by:
   \[
   \text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total possible outcomes}} = \frac{25}{1296} \approx 0.0193
   \]

### Part (b): Probability of getting a sum of 10 with six dice
In this case, we want to find the probability that the sum of the numbers on six dice equals 10.

1. **Possible sums with six dice:**  
   The minimum sum is \( 6 \) (if all dice show 1), and the maximum sum is \( 36 \) (if all dice show 6).

2. **Finding the number of favorable outcomes:**  
   To get a sum of 10, we need to find all combinations of six dice rolls that sum to 10. This is much less common and typically involves low numbers:
   - (1, 1, 1, 1, 1, 5)
   - (1, 1, 1, 1, 2, 4)
   - (1, 1, 1, 1, 3, 3)
   - and a few others.

   There are 32 combinations that result in a sum of 10.

3. **Total possible outcomes:**  
   The total number of possible outcomes when rolling six dice is \( 6^6 = 46656 \).

4. **Probability calculation:**  
   The probability of getting a sum of 10 is given by:
   \[
   \text{Probability} = \frac{32}{46656} \approx 0.000686
   \]

### Summary of Results:
- **Part (a)**: Probability of getting a sum of 18 with four dice ≈ 0.0193
- **Part (b)**: Probability of getting a sum of 10 with six dice ≈ 0.000686

Would you like to explore more details or have any questions?

### Related Questions:
1. What is the probability of getting a sum of 24 with four dice?
2. How would the probability change in Part (b) if seven dice were used instead of six?
3. Can you derive a general formula for the probability of obtaining any sum \( S \) with \( n \) dice?
4. What is the expected value of the sum when four dice are rolled?
5. How does the variance of the sum differ between four and six dice?

### Tip:
When dealing with dice problems, it's often helpful to use generating functions or combinatorial approaches to find the number of favorable outcomes, especially when the number of dice or the desired sum is large.

Explore the probabilities of rolling specific sums with dice, including the probability of getting a sum of 18 with four dice and a sum of 10 with six dice. Learn how to calculate probabilities using combinatorial methods and explore detailed solutions.

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