A dice is rolled twice ,determine the probability that number in the second roll will be higher than that in the first roll .

Options

A. `(7)/(12)`
B. `(5)/(12)`
C. `(15)/(36)`
D. `(19)/(36)`

Answer

To solve the problem of finding the probability that the number on the second roll of a die is greater than the number on the first roll, we can follow these steps:

### Step 1: Understand the Total Outcomes
When a die is rolled, it can land on any of the numbers from 1 to 6. Since the die is rolled twice, the total number of possible outcomes is calculated as follows:
- For the first roll, there are 6 possible outcomes (1, 2, 3, 4, 5, 6).
- For the second roll, there are also 6 possible outcomes.

Thus, the total number of outcomes when rolling the die twice is:
\[ \text{Total Outcomes} = 6 \times 6 = 36 \]

### Step 2: Identify Favorable Outcomes
Next, we need to determine the number of favorable outcomes where the result of the second roll is greater than the result of the first roll. We can analyze this by considering each possible outcome of the first roll:

- If the first roll is **1**, the second roll can be **2, 3, 4, 5, or 6** (5 favorable outcomes).
- If the first roll is **2**, the second roll can be **3, 4, 5, or 6** (4 favorable outcomes).
- If the first roll is **3**, the second roll can be **4, 5, or 6** (3 favorable outcomes).
- If the first roll is **4**, the second roll can be **5 or 6** (2 favorable outcomes).
- If the first roll is **5**, the second roll can be **6** (1 favorable outcome).
- If the first roll is **6**, there are no numbers greater than 6 on a die (0 favorable outcomes).

Now, we can sum the favorable outcomes:
\[ \text{Favorable Outcomes} = 5 + 4 + 3 + 2 + 1 + 0 = 15 \]

### Step 3: Calculate the Probability
The probability that the number on the second roll is greater than the number on the first roll can be calculated using the formula:
\[ \text{Probability} = \frac{\text{Number of Favorable Outcomes}}{\text{Total Outcomes}} = \frac{15}{36} \]

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

### Final Answer
Thus, the probability that the number on the second roll is higher than the number on the first roll is:
\[ \frac{5}{12} \]

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