A pair of dice is thrown.Find the following events:
(i) Same numbers on both the dice(ii)The sum is greater than 10.
(iii)Even numebr on both the dice (iv) Odd numbers on both the dice
(v) The sum is 7.

Answer

To solve the problem of finding the specified events when a pair of dice is thrown, we will analyze each event step by step.

### Step-by-Step Solution

1. **Total Outcomes When Throwing Two Dice**:
- When two dice are thrown, each die has 6 faces. Therefore, the total number of outcomes is:
\[ 6 \times 6 = 36 \]

2. **(i) Same Numbers on Both Dice**:
- The pairs where both dice show the same number are: (1,1), (2,2), (3,3), (4,4), (5,5), (6,6).
- Total outcomes for this event:
\[ \text{Total} = 6 \quad \text{(one for each number)} \]

3. **(ii) The Sum is Greater than 10**:
- The possible combinations that yield a sum greater than 10 are:
- (5,6) → 11
- (6,5) → 11
- (6,6) → 12
- Total outcomes for this event:
\[ \text{Total} = 3 \]

4. **(iii) Even Numbers on Both Dice**:
- The even numbers on a die are 2, 4, and 6. The pairs of even numbers are:
- (2,2), (2,4), (2,6)
- (4,2), (4,4), (4,6)
- (6,2), (6,4), (6,6)
- Total outcomes for this event:
\[ \text{Total} = 9 \quad \text{(3 choices for the first die and 3 for the second)} \]

5. **(iv) Odd Numbers on Both Dice**:
- The odd numbers on a die are 1, 3, and 5. The pairs of odd numbers are:
- (1,1), (1,3), (1,5)
- (3,1), (3,3), (3,5)
- (5,1), (5,3), (5,5)
- Total outcomes for this event:
\[ \text{Total} = 9 \quad \text{(3 choices for the first die and 3 for the second)} \]

6. **(v) The Sum is 7**:
- The possible combinations that yield a sum of 7 are:
- (1,6), (2,5), (3,4), (4,3), (5,2), (6,1)
- Total outcomes for this event:
\[ \text{Total} = 6 \]

### Summary of Results
- (i) Same numbers on both dice: **6**
- (ii) The sum is greater than 10: **3**
- (iii) Even numbers on both dice: **9**
- (iv) Odd numbers on both dice: **9**
- (v) The sum is 7: **6**