Q.A man is known to speak truth 3 out of 4 times. He throws a die and reports that it is a six. Find the probability that it is actually a six.

Concept: Conditional Probability (Bayes' Theorem)

Let $E_1$ be the event that the die shows a six, and $E_2$ be the event that it does not show a six.

Let $A$ be the event that the man reports a six.

Step 1: Prior probabilities

$P(E_1) = \frac{1}{6}$, $P(E_2) = \frac{5}{6}$.

Step 2: Likelihoods (probability of reporting a six given the actual outcome)

If it is a six, he tells truth: $P(A|E_1) = \frac{3}{4}$.

If it is not a six, he lies and reports a six: $P(A|E_2) = \frac{1}{4}$.

Step 3: Apply Bayes' Theorem …

Let $S$ be the event "the die shows a six" and $R$ the event "the man reports a six".

He speaks truth $3$ of $4$ times, so if it is a six he correctly reports six with probability $\tfrac34$, and if it is not a six he (lying) reports six with probability $\tfrac14$:

By Bayes' theorem, …

Method: Bayes' Theorem (finding the cause from the observed outcome)

Use this "reliability of a witness" pattern: someone reports an outcome and you want the probability the outcome is genuinely what they claim. You know how truthful the reporter is; you want $P(\text{event true}\mid\text{event reported})$.

Steps

Step 1: Set up the partition of possible causes.

List the mutually exclusive, exhaustive hypotheses (here: the die actually shows a six, or it does not) and write their prior probabilities $P(E_i)$ (these must sum to $1$).

Step 2: Write each likelihood.

For each cause, state $P(A\mid E_i)$ — the probability of the observed outcome (here: the man reports a six) under that cause.

Step 3: Get the total probability of the outcome (the denominator).

By the law of total probability,

Step 4: Apply Bayes' theorem for the cause you want. …

Common Mistakes

Mistake 1: Answering $\tfrac34$ (his truth-telling rate).

Why it's wrong: $\tfrac34$ is $P(\text{reports six}\mid\text{it is a six})$, whereas the question asks the reverse, $P(\text{it is a six}\mid\text{reports six})$. Correct approach: apply Bayes' theorem.

Mistake 2: Forgetting the "lie" branch in the denominator. …