a die is thrown twice . What is the probability that at least one of the two numbers is 4?

Answer

To find the probability that at least one of the two numbers rolled on a die is 4 when the die is thrown twice, we can follow these steps:

### Step 1: Determine the total number of outcomes
When a die is thrown twice, each die has 6 faces. Therefore, the total number of outcomes when throwing a die twice is calculated as follows:

\[ \text{Total Outcomes} = 6 \times 6 = 36 \]

### Step 2: Identify the favorable outcomes
We need to find the outcomes where at least one of the two rolls is a 4. We can list the favorable outcomes:

- If the first die is 4, the second die can be any of the 6 faces:
- (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6)

- If the second die is 4, the first die can be any of the 6 faces:
- (1, 4), (2, 4), (3, 4), (5, 4), (6, 4)

Now, we need to be careful not to double-count the outcome (4, 4), which has already been included in both lists.

Counting these, we have:
- From the first die being 4: 6 outcomes
- From the second die being 4: 5 outcomes (since (4, 4) is already counted)

Thus, the total number of favorable outcomes is:

\[ \text{Favorable Outcomes} = 6 + 5 - 1 = 10 \]

### Step 3: Calculate the probability
The probability of an event is given by the formula:

\[ P(A) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Outcomes}} \]

Substituting the values we found:

\[ P(\text{at least one 4}) = \frac{11}{36} \]

### Final Answer
The probability that at least one of the two numbers is 4 is:

\[ \frac{11}{36} \]

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