An unbiased coin is tossed is tossed a fixed number of times. If the probability of getting 4 heads equals the probability of getting 7 heads, then the probability of getting 2 heads , is

Options

A. `(55)/(2048)`
B. `(3)/(4096)`
C. `(1)/(1024)`
D. none of these

Answer

To solve the problem step by step, we will use the properties of binomial probability.

### Step-by-Step Solution:

1. **Understanding the Problem:**
We are given that the probability of getting 4 heads is equal to the probability of getting 7 heads when an unbiased coin is tossed a fixed number of times. We need to find the probability of getting 2 heads.

2. **Setting Up the Probability Formula:**
The probability of getting exactly \( r \) heads in \( n \) tosses of a coin can be expressed using the binomial probability formula:
\[ P(X = r) = \binom{n}{r} p^r q^{n-r} \]
where \( p \) is the probability of getting heads (which is \( \frac{1}{2} \)), \( q \) is the probability of getting tails (also \( \frac{1}{2} \)), and \( \binom{n}{r} \) is the binomial coefficient.

3. **Equating the Probabilities:**
We know that:
\[ P(X = 4) = P(X = 7) \]
This gives us:
\[ \binom{n}{4} \left(\frac{1}{2}\right)^4 \left(\frac{1}{2}\right)^{n-4} = \binom{n}{7} \left(\frac{1}{2}\right)^7 \left(\frac{1}{2}\right)^{n-7} \]
Simplifying this, we can cancel out \( \left(\frac{1}{2}\right)^n \) from both sides:
\[ \binom{n}{4} = \binom{n}{7} \]

4. **Using the Property of Binomial Coefficients:**
The equality \( \binom{n}{r} = \binom{n}{n-r} \) implies:
\[ 4 + 7 = n \implies n = 11 \]

5. **Finding the Probability of Getting 2 Heads:**
Now we need to find \( P(X = 2) \):
\[ P(X = 2) = \binom{11}{2} \left(\frac{1}{2}\right)^2 \left(\frac{1}{2}\right)^{11-2} \]
This simplifies to:
\[ P(X = 2) = \binom{11}{2} \left(\frac{1}{2}\right)^{11} \]

6. **Calculating the Binomial Coefficient:**
\[ \binom{11}{2} = \frac{11 \times 10}{2 \times 1} = 55 \]

7. **Final Probability Calculation:**
Therefore:
\[ P(X = 2) = 55 \cdot \left(\frac{1}{2}\right)^{11} = \frac{55}{2048} \]

### Final Answer:
The probability of getting 2 heads is \( \frac{55}{2048} \).