Two unbiased dice are thrown . Find the probability that
the total numbers on the is greater than 8 ,

Answer

To find the probability that the total numbers on two unbiased dice are greater than 8, we can follow these steps:

### Step 1: Determine the Sample Space
When two unbiased dice are thrown, each die has 6 faces. Therefore, the total number of outcomes when both dice are thrown is:
\[ \text{Total outcomes} = 6 \times 6 = 36 \]

### Step 2: Identify Favorable Outcomes
Next, we need to find the outcomes where the sum of the numbers on the two dice is greater than 8. We can list the pairs of outcomes (die1, die2) that satisfy this condition:

- For a sum of 9:
- (3, 6)
- (4, 5)
- (5, 4)
- (6, 3)

- For a sum of 10:
- (4, 6)
- (5, 5)
- (6, 4)

- For a sum of 11:
- (5, 6)
- (6, 5)

- For a sum of 12:
- (6, 6)

Now, let's count these outcomes:
- Sum of 9: 4 outcomes
- Sum of 10: 3 outcomes
- Sum of 11: 2 outcomes
- Sum of 12: 1 outcome

Adding these gives us:
\[ 4 + 3 + 2 + 1 = 10 \text{ favorable outcomes} \]

### Step 3: Calculate the Probability
The probability \( P \) of getting a sum greater than 8 is given by the ratio of the number of favorable outcomes to the total number of outcomes:
\[ P(\text{sum} > 8) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{10}{36} \]

### Step 4: Simplify the Probability
We can simplify \( \frac{10}{36} \):
\[ P(\text{sum} > 8) = \frac{5}{18} \]

### Final Answer
Thus, the probability that the total numbers on the dice are greater than 8 is:
\[ \frac{5}{18} \]

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