Dice is rolled twice. If sum of the two numbers obtained is a multiple of 4.Find the conditional probability that 4 occurs atleast once.

Answer

To solve the problem of finding the conditional probability that a 4 occurs at least once when the sum of two dice rolls is a multiple of 4, we will follow these steps:

### Step-by-Step Solution:

1. **Identify the Sample Space**:
When a die is rolled twice, the total number of outcomes is \(6 \times 6 = 36\).

2. **Determine the Relevant Sums**:
The sums that are multiples of 4 when rolling two dice are 4, 8, and 12.

3. **Calculate Outcomes for Each Relevant Sum**:
- **Sum = 4**: The pairs that give this sum are:
- (1, 3)
- (3, 1)
- (2, 2)
- Total outcomes for sum = 4: **3 outcomes**.

- **Sum = 8**: The pairs that give this sum are:
- (2, 6)
- (3, 5)
- (4, 4)
- (5, 3)
- (6, 2)
- Total outcomes for sum = 8: **5 outcomes**.

- **Sum = 12**: The only pair that gives this sum is:
- (6, 6)
- Total outcomes for sum = 12: **1 outcome**.

4. **Total Outcomes Where the Sum is a Multiple of 4**:
Adding the outcomes from the relevant sums:
\[ \text{Total outcomes} = 3 \text{ (for sum 4)} + 5 \text{ (for sum 8)} + 1 \text{ (for sum 12)} = 9 \]

5. **Identify Outcomes Where 4 Occurs at Least Once**:
- **For Sum = 4**: There are no outcomes where 4 occurs at least once.
- **For Sum = 8**: The outcome (4, 4) includes a 4.
- **For Sum = 12**: There are no outcomes where 4 occurs at least once.

Thus, the only outcome where 4 occurs at least once is from the sum of 8:
- Total outcomes where 4 occurs at least once: **1 outcome** (which is (4, 4)).

6. **Calculate the Conditional Probability**:
The conditional probability \(P(B|A)\) that 4 occurs at least once given that the sum is a multiple of 4 is given by:
\[ P(B|A) = \frac{P(A \cap B)}{P(A)} \]
Where:
- \(P(A)\) = Total outcomes where the sum is a multiple of 4 = 9.
- \(P(A \cap B)\) = Outcomes where the sum is a multiple of 4 and 4 occurs at least once = 1.

Therefore:
\[ P(B|A) = \frac{1}{9} \]

### Final Answer:
The conditional probability that 4 occurs at least once given that the sum of the two numbers obtained is a multiple of 4 is \(\frac{1}{9}\).