To solve the problem step by step, we will find the probability that the sum of the numbers on two dice is 4, given that the two numbers are different.
### Step 1: Determine the total number of outcomes when throwing two dice.
When two dice are thrown, each die has 6 faces. Therefore, the total number of outcomes is:
\[
6 \times 6 = 36
\]
**Hint:** Remember that each die is independent, so you multiply the number of outcomes for each die.
### Step 2: Define the events.
Let:
- Event A: The two numbers appearing on the dice are different.
- Event B: The sum of the numbers on the dice is 4.
**Hint:** Clearly define the events to make it easier to calculate probabilities.
### Step 3: Calculate the total number of outcomes for Event A.
To find the outcomes where the two numbers are different, we need to exclude the cases where both numbers are the same. The pairs where the numbers are the same are (1,1), (2,2), (3,3), (4,4), (5,5), and (6,6). There are 6 such outcomes.
Thus, the number of outcomes where the numbers are different is:
\[
36 - 6 = 30
\]
**Hint:** Count the outcomes where the numbers are the same and subtract from the total.
### Step 4: Calculate the number of favorable outcomes for Event B.
Now, we need to find the pairs of numbers that sum to 4. The possible pairs are:
- (1, 3)
- (2, 2)
- (3, 1)
However, since we are considering only the cases where the numbers are different, we exclude (2, 2). Therefore, the valid pairs are:
- (1, 3)
- (3, 1)
So, the number of favorable outcomes for Event B (where the sum is 4 and the numbers are different) is:
\[
2
\]
**Hint:** List all pairs that meet the criteria and remember to exclude duplicates.
### Step 5: Calculate the probability of Event A.
The probability of Event A (the two numbers are different) is:
\[
P(A) = \frac{\text{Number of favorable outcomes for A}}{\text{Total outcomes}} = \frac{30}{36} = \frac{5}{6}
\]
**Hint:** Use the formula for probability: \( P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} \).
### Step 6: Calculate the probability of Event B given Event A.
We use the formula for conditional probability:
\[
P(B|A) = \frac{P(A \cap B)}{P(A)}
\]
Where \( P(A \cap B) \) is the probability that both events occur. We already found that:
\[
P(A \cap B) = \frac{2}{36} = \frac{1}{18}
\]
Now substituting into the conditional probability formula:
\[
P(B|A) = \frac{\frac{1}{18}}{\frac{5}{6}} = \frac{1}{18} \times \frac{6}{5} = \frac{1}{15}
\]
**Hint:** Remember to simplify the fractions carefully.
### Final Answer:
The probability that the sum of the numbers on the dice is 4, given that the two numbers are different, is:
\[
\frac{1}{15}
\]
