Abhay speaks the truth only 60%. Hasan rolls a dice blindfolded and asks Abhay to tell him if the outcome is a 'prime'. Abhay says, "NO". What is the probability that the outcome is really 'prime'?

Options

A. 0.5
B. 0.75
C. 0.6
D. None of these

Answer

To solve the problem, we need to determine the probability that the outcome of the dice roll is prime given that Abhay said "NO". We can use Bayes' theorem to help us with this.

### Step-by-Step Solution:

1. **Identify the Outcomes of the Dice:**
The possible outcomes when rolling a standard six-sided die are: {1, 2, 3, 4, 5, 6}. Among these, the prime numbers are 2, 3, and 5.

2. **Determine the Probability of Rolling a Prime Number:**
There are 3 prime numbers (2, 3, 5) out of 6 possible outcomes. Therefore, the probability of rolling a prime number (P(Prime)) is:
\[ P(Prime) = \frac{3}{6} = \frac{1}{2} \]

3. **Determine the Probability of Rolling a Non-Prime Number:**
The non-prime numbers are 1, 4, and 6. Thus, the probability of rolling a non-prime number (P(Not Prime)) is:
\[ P(Not \, Prime) = \frac{3}{6} = \frac{1}{2} \]

4. **Determine the Probability of Abhay Saying "NO":**
- If the outcome is prime (which happens with probability \( \frac{1}{2} \)), Abhay tells the truth 60% of the time, meaning he will say "YES" 60% of the time and "NO" 40% of the time. Therefore, the probability that he says "NO" given that the outcome is prime (P(Say NO | Prime)) is:
\[ P(Say \, NO | Prime) = 0.4 \]
- If the outcome is not prime (which also happens with probability \( \frac{1}{2} \)), Abhay will tell the truth and say "NO" 60% of the time. Therefore, the probability that he says "NO" given that the outcome is not prime (P(Say NO | Not Prime)) is:
\[ P(Say \, NO | Not \, Prime) = 0.6 \]

5. **Calculate the Total Probability of Saying "NO":**
Using the law of total probability:
\[ P(Say \, NO) = P(Say \, NO | Prime) \cdot P(Prime) + P(Say \, NO | Not \, Prime) \cdot P(Not \, Prime) \]
Substituting the values:
\[ P(Say \, NO) = (0.4 \cdot \frac{1}{2}) + (0.6 \cdot \frac{1}{2}) = 0.2 + 0.3 = 0.5 \]

6. **Use Bayes' Theorem to Find the Probability that the Outcome is Prime Given that Abhay Said "NO":**
We want to find \( P(Prime | Say \, NO) \):
\[ P(Prime | Say \, NO) = \frac{P(Say \, NO | Prime) \cdot P(Prime)}{P(Say \, NO)} \]
Substituting the known values:
\[ P(Prime | Say \, NO) = \frac{0.4 \cdot \frac{1}{2}}{0.5} = \frac{0.2}{0.5} = 0.4 \]

### Final Answer:
The probability that the outcome is really prime given that Abhay said "NO" is **0.4**.