Two dice are throw simultaneously . Find the probability that (i) both dice show the same number, (ii) the product of the numbers on the dice is 8, (iii) the total of the numbers on the dice is greater than 9.

Answer

To solve the problem, we will break it down into three parts as specified in the question.

### Total Outcomes
When two dice are thrown, each die has 6 faces. Therefore, the total number of outcomes when throwing two dice is calculated as follows:

\[ \text{Total Outcomes} = 6 \times 6 = 36 \]

### (i) Probability that both dice show the same number
The outcomes where both dice show the same number are:
- (1, 1)
- (2, 2)
- (3, 3)
- (4, 4)
- (5, 5)
- (6, 6)

There are 6 favorable outcomes where both dice show the same number.

The probability is given by the formula:

\[ P(\text{same number}) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{6}{36} = \frac{1}{6} \]

### (ii) Probability that the product of the numbers on the dice is 8
Next, we need to find the pairs of numbers whose product is 8. The pairs that satisfy this condition are:
- (2, 4)
- (4, 2)
- (1, 8) (not possible since the maximum number on a die is 6)
- (8, 1) (not possible)
- (8, 8) (not possible)

Thus, the valid pairs are (2, 4) and (4, 2), giving us 2 favorable outcomes.

The probability is calculated as follows:

\[ P(\text{product is 8}) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{2}{36} = \frac{1}{18} \]

### (iii) Probability that the total of the numbers on the dice is greater than 9
Now, we will find the pairs of numbers that give a sum greater than 9. The possible pairs are:
- (4, 6) → 10
- (5, 5) → 10
- (5, 6) → 11
- (6, 4) → 10
- (6, 5) → 11
- (6, 6) → 12

Counting these, we find that there are 6 favorable outcomes.

The probability is given by:

\[ P(\text{sum > 9}) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{6}{36} = \frac{1}{6} \]

### Summary of Results
1. Probability that both dice show the same number: \( \frac{1}{6} \)
2. Probability that the product of the numbers on the dice is 8: \( \frac{1}{18} \)
3. Probability that the total of the numbers on the dice is greater than 9: \( \frac{1}{6} \)