15 coins are tossed. If the probability of getting at least 8 heads is equal to p, then `(8)/(p)` is equal to

Options

A. 2
B. 4
C. 8
D. 16

Answer

To solve the problem, we need to find the probability \( p \) of getting at least 8 heads when 15 coins are tossed. Then, we will calculate \( \frac{8}{p} \).

### Step-by-Step Solution:

1. **Understanding the Problem**:
We are tossing 15 coins, and we want the probability of getting at least 8 heads. This means we need to consider the cases of getting 8, 9, 10, 11, 12, 13, 14, and 15 heads.

2. **Using Binomial Distribution**:
The number of heads in 15 tosses follows a binomial distribution:
\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]
where \( n = 15 \), \( p = \frac{1}{2} \) (probability of heads), and \( k \) is the number of heads.

3. **Calculating the Probability**:
We need to calculate:
\[ P(X \geq 8) = P(X = 8) + P(X = 9) + P(X = 10) + P(X = 11) + P(X = 12) + P(X = 13) + P(X = 14) + P(X = 15) \]
This can be expressed as:
\[ P(X \geq 8) = \sum_{k=8}^{15} \binom{15}{k} \left(\frac{1}{2}\right)^{15} \]

4. **Simplifying the Expression**:
Since \( \left(\frac{1}{2}\right)^{15} \) is common in all terms, we can factor it out:
\[ P(X \geq 8) = \left(\frac{1}{2}\right)^{15} \sum_{k=8}^{15} \binom{15}{k} \]

5. **Using the Binomial Theorem**:
The sum of binomial coefficients can be simplified using the identity:
\[ \sum_{k=0}^{n} \binom{n}{k} = 2^n \]
Therefore:
\[ \sum_{k=0}^{15} \binom{15}{k} = 2^{15} \]
The sum from \( k=0 \) to \( k=7 \) can be calculated as:
\[ \sum_{k=0}^{7} \binom{15}{k} = \sum_{k=8}^{15} \binom{15}{k} \]
Thus:
\[ \sum_{k=8}^{15} \binom{15}{k} = \frac{1}{2} \cdot 2^{15} = 2^{14} \]

6. **Final Calculation of Probability**:
Now substituting back:
\[ P(X \geq 8) = \left(\frac{1}{2}\right)^{15} \cdot 2^{14} = \frac{2^{14}}{2^{15}} = \frac{1}{2} \]
Therefore, \( p = \frac{1}{2} \).

7. **Calculating \( \frac{8}{p} \)**:
Now we can find \( \frac{8}{p} \):
\[ \frac{8}{p} = \frac{8}{\frac{1}{2}} = 8 \times 2 = 16 \]

### Final Answer:
\[ \frac{8}{p} = 16 \]