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 numbers on both the dice
(iv) Odd numbers on both the dice
(v) The sum is 7.

Answer

To solve the problem of finding various events when a pair of dice is thrown, we will follow these steps:

### Step 1: Determine the Sample Space
When a pair of dice is thrown, the total number of outcomes can be calculated as follows:
- Each die has 6 faces.
- Therefore, the total number of outcomes when throwing two dice is \(6 \times 6 = 36\).

The sample space can be represented as:
\[ S = \{ (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6) \} \]

### Step 2: Event (i) - Same Numbers on Both Dice
The outcomes where both dice show the same number are:
\[ A = \{ (1,1), (2,2), (3,3), (4,4), (5,5), (6,6) \} \]
- Total outcomes for event A = 6.

### Step 3: Event (ii) - Sum Greater than 10
The outcomes where the sum is greater than 10 are:
- (5,6) → 11
- (6,5) → 11
- (6,6) → 12

Thus, the outcomes for event B are:
\[ B = \{ (5,6), (6,5), (6,6) \} \]
- Total outcomes for event B = 3.

### Step 4: Event (iii) - Even Numbers on Both Dice
The outcomes where both dice show even numbers (2, 4, 6) are:
\[ C = \{ (2,2), (2,4), (2,6), (4,2), (4,4), (4,6), (6,2), (6,4), (6,6) \} \]
- Total outcomes for event C = 9.

### Step 5: Event (iv) - Odd Numbers on Both Dice
The outcomes where both dice show odd numbers (1, 3, 5) are:
\[ D = \{ (1,1), (1,3), (1,5), (3,1), (3,3), (3,5), (5,1), (5,3), (5,5) \} \]
- Total outcomes for event D = 9.

### Step 6: Event (v) - Sum Equals 7
The outcomes where the sum equals 7 are:
- (1,6)
- (2,5)
- (3,4)
- (4,3)
- (5,2)
- (6,1)

Thus, the outcomes for event E are:
\[ E = \{ (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) \} \]
- Total outcomes for event E = 6.

### Summary of Events
- Event A (Same numbers on both dice): 6 outcomes
- Event B (Sum greater than 10): 3 outcomes
- Event C (Even numbers on both dice): 9 outcomes
- Event D (Odd numbers on both dice): 9 outcomes
- Event E (Sum equals 7): 6 outcomes