Q.Suppose you have two coins which appear identical in your pocket. You know that one is fair and one is $2$-headed. If you take one out, toss it and get a head, what is the probability that it was a fair coin?

The key idea is Conditional Probability — we update the probability of which coin was chosen based on the observed outcome (a head).

Step 1: Define events.

Let $F$ = "fair coin chosen", $T$ = "two-headed coin chosen", and $H$ = "toss shows head".

Prior: $P(F) = P(T) = \frac{1}{2}$.

Step 2: Likelihoods.

$P(H \mid F) = \frac{1}{2}$ (fair coin gives head half the time).

$P(H \mid T) = 1$ (two-headed coin always gives head).

Step 3: Apply Bayes' theorem. …

Why conditional probability is the right tool

The question asks: given that we observed a head, what’s the chance the coin was fair? That’s a classic inverse probability problem. We know the probabilities of heads if the coin is fair or two-headed, but we need to reverse the condition — from effect (head) back to cause (which coin). This is exactly what Bayes’ theorem does.

The intuition: a fair coin gives heads half the time, but a two-headed coin gives heads every time. So if we see a head, it’s more likely to have come from the two-headed coin. But we don’t know which coin we picked — each was equally likely at the start. Bayes’ theorem lets us update that initial 50–50 chance using the evidence.

Here $A$ = “coin is fair”, $B$ = “toss shows head”.

Step-by-step solution

1. Define the events clearly

   Let $F$ be the event that the chosen coin is fair.

   Let $H$ be the event that the toss shows a head.

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

2. Write down the prior probabilities

   Since the two coins look identical and you pick one at random:

3. Write down the likelihoods
   • If the coin is fair, probability of a head is $\frac{1}{2}$:

• If the coin is two-headed, probability of a head is $1$:

4. Find the total probability of getting a head By the law of total probability:

Substitute:

5. Apply Bayes’ theorem …

Method: Bayes' Theorem (reverse the conditioning)

Use Bayes when you know $P(\text{effect}\mid\text{cause})$ but want $P(\text{cause}\mid\text{effect})$ — updating a prior after seeing evidence.

Steps

Step 1: List the causes with their prior probabilities.

Here the causes are "fair coin" and "two-headed coin", each with prior $\tfrac12$.

Step 2: Write each likelihood — the chance of the observed evidence under each cause.

$P(\text{head}\mid\text{fair})=\tfrac12$, $P(\text{head}\mid\text{two-headed})=1$. …

Common Mistakes

Mistake 1: Answering $\tfrac12$ because "there are two coins".

Why it's wrong: observing a head is evidence — the two-headed coin produces heads twice as readily as the fair one — so the posterior for "fair" drops below $\tfrac12$. Correct approach: update the prior with the likelihoods via Bayes, giving $\tfrac13$.

Mistake 2: Forgetting the two-headed coin's likelihood in the denominator. …