In a simultaneous throw of a pair of dice, find the probability of getting:
an even number on one and a multiple of 3 on the other.

Answer

To find the probability of getting an even number on one die and a multiple of 3 on the other die when throwing a pair of dice, we can follow these steps:

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

### Step 2: Identify the favorable outcomes for an even number
The even numbers on a die are 2, 4, and 6. Therefore, there are 3 even numbers possible on one die.

### Step 3: Identify the favorable outcomes for a multiple of 3
The multiples of 3 on a die are 3 and 6. Therefore, there are 2 multiples of 3 possible on the other die.

### Step 4: Calculate the favorable outcomes
Now, we need to calculate the total favorable outcomes where one die shows an even number and the other die shows a multiple of 3. We can have the following combinations:

- If the first die (Die 1) shows an even number (2, 4, or 6), and the second die (Die 2) shows a multiple of 3 (3 or 6):
- Die 1 = 2, Die 2 = 3
- Die 1 = 2, Die 2 = 6
- Die 1 = 4, Die 2 = 3
- Die 1 = 4, Die 2 = 6
- Die 1 = 6, Die 2 = 3
- Die 1 = 6, Die 2 = 6

This gives us a total of \(3 \text{ (even numbers)} \times 2 \text{ (multiples of 3)} = 6\) favorable outcomes.

### Step 5: Calculate the probability
Now we can calculate the probability using the formula:
\[ \text{Probability} = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}} = \frac{6}{36} = \frac{1}{6} \]

### Final Answer
The probability of getting an even number on one die and a multiple of 3 on the other die is:
\[ \frac{1}{6} \]

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