A die is thrown twice. Find the probability that:
5 will come up at least once

Answer

To find the probability that a 5 will come up at least once when a die is thrown twice, we can follow these steps:

### Step 1: Determine the Sample Space
When a die is thrown twice, the total number of outcomes can be calculated as follows:
- Each die has 6 faces, so when thrown twice, the total number of outcomes is:
\[ 6 \times 6 = 36 \]
Thus, the sample space consists of 36 possible outcomes.

### Step 2: Calculate the Favorable Outcomes
Next, we need to find the number of favorable outcomes where at least one of the throws results in a 5.

We can approach this by calculating the complementary probability (the probability of not getting a 5 at all) and then subtracting it from 1.

1. **Calculate the probability of not getting a 5 in a single throw**:
- The outcomes that are not 5 are: 1, 2, 3, 4, 6 (5 outcomes).
- Therefore, the probability of not getting a 5 in one throw is:
\[ P(\text{not 5}) = \frac{5}{6} \]

2. **Calculate the probability of not getting a 5 in two throws**:
- Since the throws are independent, the probability of not getting a 5 in both throws is:
\[ P(\text{not 5 in two throws}) = P(\text{not 5}) \times P(\text{not 5}) = \left(\frac{5}{6}\right) \times \left(\frac{5}{6}\right) = \frac{25}{36} \]

3. **Calculate the probability of getting at least one 5**:
- The probability of getting at least one 5 in two throws is:
\[ P(\text{at least one 5}) = 1 - P(\text{not 5 in two throws}) = 1 - \frac{25}{36} = \frac{11}{36} \]

### Step 3: Conclusion
Thus, the probability that a 5 will come up at least once when a die is thrown twice is:
\[ \frac{11}{36} \]