Two dice are thrown simultaneously. What is the probability of obtaining a multiple of 2 on one of them and a multiple of 3 on the other

Options

A. `5/36`
B. `11/36`
C. `1/6`
D. `1/3`

Answer

To solve the problem of finding the probability of obtaining a multiple of 2 on one die and a multiple of 3 on the other when two dice are thrown, we can follow these steps:

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

### Step 2: Identify the multiples of 2 and 3
- The multiples of 2 on a die are: 2, 4, and 6.
- The multiples of 3 on a die are: 3 and 6.

### Step 3: Calculate the favorable outcomes
We need to find the combinations where one die shows a multiple of 2 and the other die shows a multiple of 3.

1. **Case 1**: First die is a multiple of 2, second die is a multiple of 3.
- If the first die shows 2: The second die can show 3 or 6 → (2, 3), (2, 6)
- If the first die shows 4: The second die can show 3 or 6 → (4, 3), (4, 6)
- If the first die shows 6: The second die can show 3 or 6 → (6, 3), (6, 6)

So, from this case, we have the outcomes:
- (2, 3), (2, 6)
- (4, 3), (4, 6)
- (6, 3), (6, 6)

Total from Case 1 = 6 outcomes.

2. **Case 2**: First die is a multiple of 3, second die is a multiple of 2.
- If the first die shows 3: The second die can show 2, 4, or 6 → (3, 2), (3, 4), (3, 6)
- If the first die shows 6: The second die can show 2, 4, or 6 → (6, 2), (6, 4), (6, 6)

So, from this case, we have the outcomes:
- (3, 2), (3, 4), (3, 6)
- (6, 2), (6, 4), (6, 6)

Total from Case 2 = 6 outcomes.

### Step 4: Combine the outcomes
Now, we combine the outcomes from both cases:
- From Case 1: (2, 3), (2, 6), (4, 3), (4, 6), (6, 3), (6, 6) → 6 outcomes
- From Case 2: (3, 2), (3, 4), (3, 6), (6, 2), (6, 4), (6, 6) → 6 outcomes

However, we notice that (6, 6) is counted in both cases, so we need to subtract this duplicate outcome.

Total favorable outcomes = 6 + 6 - 1 = 11

### Step 5: Calculate the probability
The probability is given by the ratio of favorable outcomes to total outcomes:
\[ \text{Probability} = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{11}{36} \]

Thus, the final answer is:
\[ \text{Probability} = \frac{11}{36} \]