A dice is rolled three times and sum of three numbers appearing on the uppermost face is 15. The chance that the first roll was four is

Options

A. `2//5`
B. `1//5`
C. `1//6`
D. None of these

Answer

To solve the problem step by step, we need to find the probability that the first roll of a die is 4, given that the sum of three rolls is 15.

### Step 1: Understanding the Problem
We need to find the probability that the first roll is 4, given that the total of three rolls equals 15.

### Step 2: Total Outcomes for Three Rolls
When a die is rolled three times, the total number of possible outcomes is:
\[ 6 \times 6 \times 6 = 216 \]

### Step 3: Finding Favorable Outcomes
We need to find the number of favorable outcomes where the first roll is 4 and the sum of all three rolls is 15.

Let the three rolls be \(X_1\), \(X_2\), and \(X_3\). We know:
- \(X_1 = 4\)
- \(X_2 + X_3 = 15 - 4 = 11\)

### Step 4: Possible Values for \(X_2\) and \(X_3\)
Now we need to find pairs \((X_2, X_3)\) such that:
\[ X_2 + X_3 = 11 \]
Both \(X_2\) and \(X_3\) must be between 1 and 6 (the possible outcomes of a die).

The possible pairs that satisfy \(X_2 + X_3 = 11\) are:
- \(X_2 = 5\), \(X_3 = 6\)
- \(X_2 = 6\), \(X_3 = 5\)

Thus, there are **2 favorable outcomes** where the first roll is 4.

### Step 5: Finding Total Outcomes for Sum = 15
Next, we need to find the total number of outcomes where the sum of the three rolls equals 15. We can do this by considering all possible combinations of rolls that add up to 15.

To find the combinations, we can use a systematic approach or enumeration. After checking all combinations, we find the total number of combinations that yield a sum of 15 is **10**.

### Step 6: Calculating the Probability
Now we can calculate the probability that the first roll is 4 given that the sum is 15:
\[ P(X_1 = 4 | X_1 + X_2 + X_3 = 15) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes for sum = 15}} = \frac{2}{10} = \frac{1}{5} \]

### Final Answer
The probability that the first roll was 4, given that the sum of the three numbers is 15, is:
\[ \boxed{\frac{1}{5}} \]