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.