To find the probability that the numbers shown on two throws of a die differ by 2, we can follow these steps:
### Step 1: Determine the total number of outcomes
When a die is thrown twice, each throw has 6 possible outcomes (1 to 6). Therefore, the total number of outcomes when throwing the die twice is:
\[
6 \times 6 = 36
\]
### Step 2: Identify the favorable outcomes
Next, we need to find the pairs of outcomes where the numbers differ by 2. We can list these pairs systematically:
1. If the first throw is 1, the second throw can be 3 (1, 3).
2. If the first throw is 2, the second throw can be 4 (2, 4).
3. If the first throw is 3, the second throw can be 1 or 5 (3, 1) and (3, 5).
4. If the first throw is 4, the second throw can be 2 or 6 (4, 2) and (4, 6).
5. If the first throw is 5, the second throw can be 3 (5, 3).
6. If the first throw is 6, the second throw can be 4 (6, 4).
Now, let's list all the pairs:
- (1, 3)
- (3, 1)
- (2, 4)
- (4, 2)
- (3, 5)
- (5, 3)
- (4, 6)
- (6, 4)
Counting these pairs, we find there are a total of 8 favorable outcomes.
### Step 3: Calculate the probability
The probability \( P \) that the numbers shown differ by 2 can be calculated using the formula:
\[
P = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{8}{36}
\]
### Step 4: Simplify the probability
We can simplify \( \frac{8}{36} \) by dividing both the numerator and the denominator by their greatest common divisor, which is 4:
\[
P = \frac{8 \div 4}{36 \div 4} = \frac{2}{9}
\]
Thus, the probability that the numbers shown up differ by 2 is \( \frac{2}{9} \).
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