Question

A bag contains 2n + 1 coins. It is known that n of these coins have head on both sides whereas the other n + 1 coins are fair. One coin is selected at random and tossed. If the probability that toss results in heads is $\frac{31}{42}$, then the value of n is

• A. 8
• B. 5
• C. 10
• D. 6

Hint:

When dealing with probabilities in problems involving conditional events, remember to use the law of total probability. In this case, break down the probability of heads into two cases: selecting a double-headed coin and selecting a fair coin, and then sum the probabilities.

Answer 1

The Correct Option is C

The correct answer is: (C): 10

We are given a bag containing $2n + 1$ coins. Out of these, $n$ coins have heads on both sides (double-headed), and the remaining $n + 1$ coins are fair. One coin is selected at random and tossed. The probability that the toss results in heads is $\frac{31}{42}$. We are tasked with finding the value of $n$.

Step 1: Probability of selecting a coin

The probability of selecting a double-headed coin is $$\frac{n}{2n + 1}$$ and the probability of selecting a fair coin is $$\frac{n + 1}{2n + 1}.$$

Step 2: Probability of getting heads

For a double-headed coin, the probability of getting heads is 1. For a fair coin, the probability of getting heads is $$\frac{1}{2}.$$

Step 3: Total probability of getting heads

By the law of total probability: $$P(\text{Heads}) = \frac{n}{2n + 1} \times 1 + \frac{n + 1}{2n + 1} \times \frac{1}{2}$$ Simplifying: $$P(\text{Heads}) = \frac{n}{2n + 1} + \frac{n + 1}{2(2n + 1)}$$

Step 4: Equating to the given probability

Since $P(\text{Heads}) = \frac{31}{42}$, we equate: $$\frac{n}{2n + 1} + \frac{n + 1}{2(2n + 1)} = \frac{31}{42}$$ Combining like terms: $$\frac{2n + (n + 1)}{2(2n + 1)} = \frac{31}{42}$$ This simplifies to: $$\frac{3n + 1}{2(2n + 1)} = \frac{31}{42}$$

Step 5: Solve for $n$

Cross-multiplying: $$42(3n + 1) = 62(2n + 1)$$ Expanding: $$126n + 42 = 124n + 62$$ Simplifying: $$2n = 20 \quad \Rightarrow \quad n = 10$$

Conclusion:

The value of $n$ is 10, so the correct answer is (C): 10.