Q.Probability that A speaks truth is $\frac{4}{5}$. A coin is tossed. A reports that a head appears. The probability that actually there was head is (A) $\frac{4}{5}$ (B) $\frac{1}{2}$ (C) $\frac{1}{5}$ (D) $\frac{2}{5}$

Concept: Conditional Probability (Bayes' Theorem)

We need $P(\text{Head} \mid \text{A reports head})$.

Let $H$ = head occurs, $T$ = tail occurs, $R$ = A reports head.

Given: $P(H) = \frac{1}{2}$, $P(T) = \frac{1}{2}$, $P(R \mid H) = \frac{4}{5}$ (truth), $P(R \mid T) = \frac{1}{5}$ (lie).

By Bayes' theorem: …

Why Bayes’ theorem?

The problem gives us the probability that A speaks the truth ($\frac{4}{5}$), and we know the coin is fair. But the question asks: given that A reports a head, what is the chance the coin actually showed a head? That’s a classic “inverse probability” situation — we need to update our belief about the coin toss using the report we heard. Bayes’ theorem is the tool for exactly this.

Let’s define the events clearly:

• $H$: the coin shows a head.
• $T$: the coin shows a tail.
• $R_H$: A reports that a head appears.

We know:

• $P(H) = P(T) = \frac{1}{2}$ (fair coin).
• $P(\text{A speaks truth}) = \frac{4}{5}$. So if the coin is head, A reports head with probability $\frac{4}{5}$; if the coin is tail, A reports tail with probability $\frac{4}{5}$.
• If A lies (probability $\frac{1}{5}$), then when the coin is head, A reports tail; when the coin is tail, A reports head.

We want $P(H \mid R_H)$.

1. Identify the probabilities of reporting a head in each case.

   • If the coin is head ($H$): A tells truth → reports head: probability $\frac{4}{5}$. A lies → reports tail: probability $\frac{1}{5}$. So $P(R_H \mid H) = \frac{4}{5}$.
   • If the coin is tail ($T$): A tells truth → reports tail: probability $\frac{4}{5}$. A lies → reports head: probability $\frac{1}{5}$. So $P(R_H \mid T) = \frac{1}{5}$.

2. Apply Bayes’ theorem.

3. Plug in the numbers. …

Method: Bayes' theorem for a reliability / witness problem

Use this when a source of known reliability reports an outcome and you must find the probability the outcome truly occurred, given the report.

Steps

Step 1: Fix the prior and the reliability.

Write the prior probabilities of the true states (a fair coin gives $P(H)=P(T)=\tfrac12$) and the reliability $p$ that the source reports truthfully; it lies with probability $1-p$.

Step 2: Turn reliability into likelihoods of the report. …

Common Mistakes

Mistake 1: Answering $\tfrac12$ because the coin is fair.

Why it's wrong: A's report is extra information; since A tells the truth more often than not, a reported head genuinely raises the chance the coin was a head. Correct approach: update with Bayes' theorem, which gives $\tfrac45$.

Mistake 2: Setting $P(\text{reports head}\mid \text{tail})$ to $0$. …