A sample contains large number of nuclei. The probability that a nucleus in sample will decay after four half lives is

Options

A. (a)`(1)/(4)`
B. (b)`(3)/(4)`
C. (c)`(15)/(16)`
D. (d)`(7)/(16)`

Answer

To find the probability that a nucleus in a sample will decay after four half-lives, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Concept of Half-Life**:
The half-life of a radioactive substance is the time required for half of the nuclei in a sample to decay. After one half-life, half of the original nuclei remain.

2. **Formula for Remaining Nuclei**:
The number of nuclei remaining after \( n \) half-lives can be expressed as:
\[ A = A_0 \left( \frac{1}{2} \right)^n \]
where:
- \( A_0 \) is the initial number of nuclei,
- \( A \) is the number of nuclei remaining after \( n \) half-lives.

3. **Applying the Formula for Four Half-Lives**:
For \( n = 4 \):
\[ A = A_0 \left( \frac{1}{2} \right)^4 = A_0 \left( \frac{1}{16} \right) \]

4. **Calculating the Probability of Decay**:
The probability \( P \) that a nucleus will decay is given by:
\[ P = 1 - \frac{A}{A_0} \]
Substituting the value of \( A \):
\[ P = 1 - \frac{A_0 \left( \frac{1}{16} \right)}{A_0} \]
Simplifying this gives:
\[ P = 1 - \frac{1}{16} \]

5. **Final Calculation**:
\[ P = 1 - \frac{1}{16} = \frac{16 - 1}{16} = \frac{15}{16} \]

### Conclusion:
The probability that a nucleus in the sample will decay after four half-lives is \( \frac{15}{16} \).