A fair coin is tossed a fixed number times . If the probability of getting seven heads is equal to that of getting nine heads then the probability of getting two heads is

Answer

To solve the problem, we need to find the probability of getting 2 heads when a fair coin is tossed a fixed number of times, given that the probability of getting 7 heads is equal to that of getting 9 heads.

### Step-by-Step Solution:

1. **Define the Number of Tosses**:
Let the number of times the coin is tossed be \( n \).

2. **Probability of Getting 7 Heads**:
The probability of getting exactly 7 heads in \( n \) tosses can be expressed using the binomial distribution formula:
\[ P(X = 7) = \binom{n}{7} \left(\frac{1}{2}\right)^7 \left(\frac{1}{2}\right)^{n-7} = \binom{n}{7} \left(\frac{1}{2}\right)^n \]

3. **Probability of Getting 9 Heads**:
Similarly, the probability of getting exactly 9 heads is:
\[ P(X = 9) = \binom{n}{9} \left(\frac{1}{2}\right)^9 \left(\frac{1}{2}\right)^{n-9} = \binom{n}{9} \left(\frac{1}{2}\right)^n \]

4. **Set the Probabilities Equal**:
According to the problem, the probability of getting 7 heads is equal to the probability of getting 9 heads:
\[ \binom{n}{7} \left(\frac{1}{2}\right)^n = \binom{n}{9} \left(\frac{1}{2}\right)^n \]
Since \(\left(\frac{1}{2}\right)^n\) is common on both sides, we can cancel it out:
\[ \binom{n}{7} = \binom{n}{9} \]

5. **Using the Property of Binomial Coefficients**:
The equality \(\binom{n}{7} = \binom{n}{9}\) implies:
\[ n - 7 = 9 \quad \text{or} \quad n = 16 \]

6. **Calculate the Probability of Getting 2 Heads**:
Now we need to find the probability of getting exactly 2 heads when \( n = 16 \):
\[ P(X = 2) = \binom{16}{2} \left(\frac{1}{2}\right)^2 \left(\frac{1}{2}\right)^{16-2} = \binom{16}{2} \left(\frac{1}{2}\right)^{16} \]

7. **Calculate \(\binom{16}{2}\)**:
\[ \binom{16}{2} = \frac{16!}{2!(16-2)!} = \frac{16 \times 15}{2 \times 1} = 120 \]

8. **Final Probability Calculation**:
Now substituting back:
\[ P(X = 2) = 120 \left(\frac{1}{2}\right)^{16} = \frac{120}{65536} = \frac{15}{8192} \]

### Final Answer:
The probability of getting 2 heads is:
\[ \frac{15}{8192} \]