When two dice are thrown simultaneously, then what is the probability that there is exactly one `6`?

Options

A. `(1)/(9)`
B. `(1)/(6)`
C. `(6)/(23)`
D. `(2)/(9)`

Answer

To find the probability of getting exactly one `6` when two dice are thrown, we can follow these steps:

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

### Step 2: Identify the Favorable Outcomes
Next, we need to find the outcomes where exactly one die shows a `6`. This can happen in two scenarios:
1. The first die shows `6` and the second die shows any number from `1` to `5`.
2. The first die shows any number from `1` to `5` and the second die shows `6`.

Let's list the favorable outcomes:
- If the first die is `6`, the second die can be `1`, `2`, `3`, `4`, or `5`:
- (6, 1)
- (6, 2)
- (6, 3)
- (6, 4)
- (6, 5)

This gives us **5 outcomes**.

- If the second die is `6`, the first die can be `1`, `2`, `3`, `4`, or `5`:
- (1, 6)
- (2, 6)
- (3, 6)
- (4, 6)
- (5, 6)

This also gives us **5 outcomes**.

Adding these together, the total number of favorable outcomes for exactly one `6` is:
\[ 5 + 5 = 10 \]

### Step 3: Calculate the Probability
Now, we can calculate the probability of getting exactly one `6` using the formula:
\[ \text{Probability} = \frac{\text{Number of Favorable Outcomes}}{\text{Total Outcomes}} = \frac{10}{36} \]

### Step 4: Simplify the Probability
We can simplify the fraction:
\[ \frac{10}{36} = \frac{5}{18} \]

### Final Answer
Thus, the probability of getting exactly one `6` when two dice are thrown is:
\[ \frac{5}{18} \]

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