A pair of fair dice is rolled. What is the probability that the second dice lands on a higher value than does the first?

Options

A. `(1)/(4)`
B. `(1)/(6)`
C. `(5)/(12)`
D. `(5)/(18)`

Answer

To solve the problem of finding the probability that the second die lands on a higher value than the first die when a pair of fair dice is rolled, we can follow these steps:

### Step-by-Step Solution:

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

**Hint**: Remember that each die operates independently, so you multiply the number of outcomes for each die.

2. **Count Favorable Outcomes**:
We need to count the number of outcomes where the second die shows a value higher than the first die. We can analyze this based on the value shown on the first die:

- If the first die shows **1**: The second die can show **2, 3, 4, 5, or 6** (5 possibilities).
- If the first die shows **2**: The second die can show **3, 4, 5, or 6** (4 possibilities).
- If the first die shows **3**: The second die can show **4, 5, or 6** (3 possibilities).
- If the first die shows **4**: The second die can show **5 or 6** (2 possibilities).
- If the first die shows **5**: The second die can show **6** (1 possibility).
- If the first die shows **6**: The second die cannot show a higher value (0 possibilities).

Now, we sum these possibilities:
\[ \text{Favorable Outcomes} = 5 + 4 + 3 + 2 + 1 + 0 = 15 \]

**Hint**: Break down the counting based on the value of the first die to ensure you account for all possibilities.

3. **Calculate the Probability**:
The probability that the second die lands on a higher value than the first die is given by the ratio of favorable outcomes to total outcomes:
\[ \text{Probability} = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}} = \frac{15}{36} \]

4. **Simplify the Probability**:
We can simplify this fraction:
\[ \frac{15}{36} = \frac{5}{12} \]

**Hint**: Always check if the fraction can be simplified by finding the greatest common divisor (GCD).

### Final Answer:
The probability that the second die lands on a higher value than the first die is:
\[ \frac{5}{12} \]