A bag contains n+1 coins, one of which is a two heads coin and the rest are fair. A coin is selected at random and tossed. If the probability. That the toss result in a head is 7/12 then the value of n is

Answer

To solve the problem step by step, we will break down the information given and use it to find the value of \( n \).

### Step 1: Understand the Problem
We have a bag containing \( n + 1 \) coins:
- 1 coin is a two-headed coin (biased).
- \( n \) coins are fair coins (each having one head and one tail).

### Step 2: Define Probabilities
- The probability of getting a head from the two-headed coin is \( 1 \).
- The probability of getting a head from a fair coin is \( \frac{1}{2} \).

### Step 3: Calculate the Total Probability of Getting a Head
The total probability of getting a head when a coin is selected at random can be calculated using the Law of Total Probability:

\[ P(\text{Head}) = P(\text{Choosing two-headed coin}) \times P(\text{Head | two-headed coin}) + P(\text{Choosing fair coin}) \times P(\text{Head | fair coin}) \]

- The probability of choosing the two-headed coin is \( \frac{1}{n + 1} \).
- The probability of choosing a fair coin is \( \frac{n}{n + 1} \).

Thus, we can write:

\[ P(\text{Head}) = \left(\frac{1}{n + 1} \times 1\right) + \left(\frac{n}{n + 1} \times \frac{1}{2}\right) \]

### Step 4: Simplify the Expression
Now, substituting the probabilities into the equation:

\[ P(\text{Head}) = \frac{1}{n + 1} + \frac{n}{2(n + 1)} \]

To combine these fractions, we need a common denominator:

\[ P(\text{Head}) = \frac{2}{2(n + 1)} + \frac{n}{2(n + 1)} = \frac{2 + n}{2(n + 1)} \]

### Step 5: Set the Probability Equal to Given Value
According to the problem, this probability is given as \( \frac{7}{12} \):

\[ \frac{2 + n}{2(n + 1)} = \frac{7}{12} \]

### Step 6: Cross-Multiply to Solve for \( n \)
Cross-multiplying gives:

\[ 12(2 + n) = 7 \cdot 2(n + 1) \]

Expanding both sides:

\[ 24 + 12n = 14n + 14 \]

### Step 7: Rearranging the Equation
Now, rearranging the equation to isolate \( n \):

\[ 24 - 14 = 14n - 12n \]

This simplifies to:

\[ 10 = 2n \]

### Step 8: Solve for \( n \)
Dividing both sides by 2 gives:

\[ n = 5 \]

### Final Answer
The value of \( n \) is \( 5 \).

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