If three fair dice are thrown and the sum is an odd number, then the probability that all the three dice show an odd number is

Options

A. `(3)/(4)`
B. `(1)/(2)`
C. `(5)/(6)`
D. `(1)/(4)`

Answer

To solve the problem, we need to find the probability that all three dice show an odd number given that the sum of the numbers on the three dice is odd.

### Step-by-Step Solution:

1. **Understanding the Conditions**:
When three dice are thrown, the possible outcomes for each die are 1, 2, 3, 4, 5, and 6. The odd numbers are 1, 3, and 5, while the even numbers are 2, 4, and 6. The sum of the numbers on the dice can be odd or even.

2. **Identifying Cases for Odd Sum**:
The sum of three numbers is odd if:
- All three numbers are odd (O, O, O)
- One number is odd and two numbers are even (O, E, E)

The case where two numbers are odd and one is even (O, O, E) is not possible because the sum of two odd numbers is even, and adding an even number would still result in an even sum.

3. **Calculating Total Outcomes for Odd Sum**:
- **Case 1**: All three dice show odd numbers (O, O, O).
- Each die can show one of 3 odd numbers (1, 3, 5).
- Total outcomes for this case = \(3 \times 3 \times 3 = 27\).

- **Case 2**: One die shows an odd number and two dice show even numbers (O, E, E).
- Choose which die shows the odd number (3 ways to choose which die is odd).
- The odd die can show one of 3 odd numbers (1, 3, 5).
- The two even dice can each show one of 3 even numbers (2, 4, 6).
- Total outcomes for this case = \(3 \times 3 \times 3 \times 3 = 81\) (3 ways to choose the odd die, 3 choices for the odd number, and 3 choices for each of the two even numbers).

- **Total Outcomes for Odd Sum**:
\[ \text{Total outcomes} = 27 \text{ (all odd)} + 81 \text{ (one odd, two even)} = 108 \]

4. **Calculating Favorable Outcomes**:
The only favorable outcome for our event (all three dice showing odd numbers) is the case where all three dice are odd, which we calculated as 27.

5. **Calculating Probability**:
The probability \(P\) that all three dice show an odd number given that the sum is odd is given by:
\[ P = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes for odd sum}} = \frac{27}{108} = \frac{1}{4} \]

### Final Answer:
Thus, the probability that all three dice show an odd number given that the sum is odd is \(\frac{1}{4}\).