Question:

\textbf{Step 1: Define Events} Let $H$ be the event that a head actually appears on the coin. Let $T$ be the event that a tail actually appears on the coin. Let $R$ be the event that A reports a head. The probability that A speaks the truth is $P(\text{A speaks truth}) = 4/5$. The probability that A lies is $P(\text{A lies}) = 1 - 4/5 = 1/5$. \textbf{Step 2: Set up Prior Probabilities} Since the coin is fair, the prior probabilities are: $P(H) = 1/2$ $P(T) = 1/2$ \textbf{Step 3: Determine Conditional Probabilities for A's Report} The probability that A reports a head given that a head actually appeared is the probability that A speaks the truth: $P(R|H) = P(\text{A speaks truth}) = 4/5$. The probability that A reports a head given that a tail actually appeared is the probability that A lies: $P(R|T) = P(\text{A lies}) = 1/5$. \textbf{Step 4: Calculate the Total Probability of A Reporting a Head} Using the Law of Total Probability: $$P(R) = P(R|H)P(H) + P(R|T)P(T)$$ $$P(R) = \left(\frac{4}{5}\right)\left(\frac{1}{2}\right) + \left(\frac{1}{5}\right)\left(\frac{1}{2}\right)$$ $$P(R) = \frac{4}{10} + \frac{1}{10}$$ $$P(R) = \frac{5}{10} = \frac{1}{2}$$ \textbf{Step 5: Apply Bayes' Theorem to Find the Probability of an Actual Head Given A Reports a Head} We want to find $P(H|R)$, the probability that there was actually a head given that A reports a head. Using Bayes' Theorem: $$P(H|R) = \frac{P(R|H)P(H)}{P(R)}$$ $$P(H|R) = \frac{\left(\frac{4}{5}\right)\left(\frac{1}{2}\right)}{\frac{1}{2}}$$ $$P(H|R) = \frac{\frac{4}{10}}{\frac{5}{10}}$$ $$P(H|R) = \frac{4}{5}$$