Question

A bag contains \( (n + 1) \) coins. It is known that one of these coins has a head on both sides, whereas the other coins are fair. One of these coins is selected at random and tossed. If the probability that the toss results in heads is \( \frac{7}{12} \), then the value of \( n \) is :

• A. 5
• B. 3
• C. 2
• D. 4

Hint:

When solving probability problems, carefully account for the different cases and use the law of total probability to combine the outcomes.

Answer 1

The Correct Option is A

Step 1: Understand the problem setup.
The bag contains \( (n + 1) \) coins, one of which is double-headed, and the others are fair coins. A coin is selected at random and tossed, and we are asked to find the value of \( n \) such that the probability of getting heads is \( \frac{7}{12} \).

Step 2: Calculate the probabilities.
- The probability of selecting the double-headed coin is \( \frac{1}{n+1} \), and the probability of getting heads with this coin is 1.
- The probability of selecting a fair coin is \( \frac{n}{n+1} \), and the probability of getting heads with a fair coin is \( \frac{1}{2} \).

Step 3: Set up the equation.
The total probability of getting heads is the weighted sum of the probabilities of heads for each type of coin:
\[ P(\text{Heads}) = \frac{1}{n+1} \times 1 + \frac{n}{n+1} \times \frac{1}{2} \]
Simplifying the equation:
\[ P(\text{Heads}) = \frac{1}{n+1} + \frac{n}{2(n+1)} = \frac{2 + n}{2(n+1)} \]

Step 4: Use the given probability.
We are told that the probability of getting heads is \( \frac{7}{12} \), so we set the equation equal to \( \frac{7}{12} \):
\[ \frac{2 + n}{2(n+1)} = \frac{7}{12} \]

Step 5: Solve for \( n \).
Cross-multiply to solve for \( n \):
\[ 12(2 + n) = 7 \times 2(n+1) \]
Expanding both sides: \[ 24 + 12n = 14n + 14 \]
Simplifying the equation: \[ 24 - 14 = 14n - 12n \] \[ 10 = 2n \] \[ n = 5 \]

Step 6: Conclusion.
The value of \( n \) is 5. Therefore, the correct answer is option (A).