Probability that A speaks truth is `4/5` . A coin is tossed. A reports that a head appears. The probability that actually there was head is (A) `4/5` (B)`1/2` (C) `1/5` (D) `2/5`

Explanation

To solve the problem, we need to find the probability that there was actually a head when A reports that a head appears. We can use Bayes' theorem to find this probability.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Probability that A speaks the truth (P(T)) = \( \frac{4}{5} \)
- Probability that A speaks falsely (P(F)) = \( 1 - P(T) = 1 - \frac{4}{5} = \frac{1}{5} \)
- Probability of getting heads when a coin is tossed (P(H)) = \( \frac{1}{2} \)
- Probability of getting tails when a coin is tossed (P(Tail)) = \( \frac{1}{2} \)

2. **Define the Required Probability:**
- We want to find the probability that there was actually a head given that A reports a head (P(H | Reported H)).

3. **Apply Bayes' Theorem:**
- According to Bayes' theorem:
\[ P(H | \text{Reported H}) = \frac{P(\text{Reported H} | H) \cdot P(H)}{P(\text{Reported H})} \]

4. **Calculate P(Reported H | H):**
- If there is a head, A speaks the truth, so:
\[ P(\text{Reported H} | H) = P(T) = \frac{4}{5} \]

5. **Calculate P(Reported H | T):**
- If there is a tail, A speaks falsely, so:
\[ P(\text{Reported H} | T) = P(F) = \frac{1}{5} \]

6. **Calculate P(Reported H):**
- Using the law of total probability:
\[ P(\text{Reported H}) = P(\text{Reported H} | H) \cdot P(H) + P(\text{Reported H} | T) \cdot P(T) \]
- Substitute the values:
\[ P(\text{Reported H}) = \left(\frac{4}{5} \cdot \frac{1}{2}\right) + \left(\frac{1}{5} \cdot \frac{1}{2}\right) \]
\[ = \frac{4}{10} + \frac{1}{10} = \frac{5}{10} = \frac{1}{2} \]

7. **Substitute Back into Bayes' Theorem:**
- Now we can substitute back into Bayes' theorem:
\[ P(H | \text{Reported H}) = \frac{P(\text{Reported H} | H) \cdot P(H)}{P(\text{Reported H})} \]
\[ = \frac{\left(\frac{4}{5}\right) \cdot \left(\frac{1}{2}\right)}{\frac{1}{2}} \]
\[ = \frac{4}{5} \]

8. **Final Answer:**
- Therefore, the probability that there was actually a head when A reports a head is \( \frac{4}{5} \).

### Conclusion:
The correct option is (A) \( \frac{4}{5} \).