JEE Mathematics 2025 Question with Solution

Given: A bag contains $19$ unbiased coins and $1$ coin with heads on both sides. A coin is drawn and tossed, and heads turns up.

Find: If $P(\text{Unbiased} \mid \text{Heads}) = \frac{m}{n}$ with $\gcd(m,n)=1$, find $n^2-m^2$.

Let $A$ be the event that the drawn coin is unbiased and $B$ be the event that heads turns up.

Then

$$P(A)=\frac{19}{20}, \qquad P(A^c)=\frac{1}{20}$$

and

$$P(B\mid A)=\frac{1}{2}, \qquad P(B\mid A^c)=1$$

Using total probability,

$$P(B)=P(B\mid A)P(A)+P(B\mid A^c)P(A^c)$$ $$P(B)=\left(\frac{1}{2}\right)\left(\frac{19}{20}\right)+1\left(\frac{1}{20}\right)$$ $$P(B)=\frac{19}{40}+\frac{1}{20}=\frac{19}{40}+\frac{2}{40}=\frac{21}{40}$$

Now by Bayes' theorem,

$$P(A\mid B)=\frac{P(B\mid A)P(A)}{P(B)}$$ $$P(A\mid B)=\frac{\left(\frac{1}{2}\right)\left(\frac{19}{20}\right)}{\frac{21}{40}}=\frac{\frac{19}{40}}{\frac{21}{40}}=\frac{19}{21}$$

So, $m=19$ and $n=21$.

Therefore,

$$n^2-m^2=21^2-19^2=(21+19)(21-19)=40\times 2=80$$

Hence, the correct option is A.

Given: There are $20$ coins in total: $19$ unbiased coins and $1$ two-headed coin.

Find: The value of $n^2-m^2$ when the probability that the chosen coin was unbiased, given that heads appeared, is $\frac{m}{n}$.

This is a conditional probability problem, so Bayes' theorem applies.

1. Define events:

• $A$: the drawn coin is unbiased
• $B$: heads turns up

2. Compute the basic probabilities:

$$P(A)=\frac{19}{20}, \qquad P(A^c)=\frac{1}{20}$$ $$P(B\mid A)=\frac{1}{2}, \qquad P(B\mid A^c)=1$$

3. Find the probability of getting heads:

$$P(B)=P(B\mid A)P(A)+P(B\mid A^c)P(A^c)$$ $$P(B)=\frac{1}{2}\cdot\frac{19}{20}+1\cdot\frac{1}{20}$$ $$P(B)=\frac{19}{40}+\frac{1}{20}=\frac{21}{40}$$

4. Apply Bayes' theorem:

$$P(A\mid B)=\frac{P(B\mid A)P(A)}{P(B)}$$ $$P(A\mid B)=\frac{\frac{1}{2}\cdot\frac{19}{20}}{\frac{21}{40}}=\frac{19}{21}$$

Thus, $\frac{m}{n}=\frac{19}{21}$, so $m=19$ and $n=21$.

5. Evaluate the required expression:

$$21^2-19^2=(21+19)(21-19)=40\cdot 2=80$$

Therefore, the answer is $80$, so the correct option is A.