A sample contains a large number of nuclei. The probability that a nucleus in the sample will decay after four half - lives is `(a)/(b)` where a and b are least positive integers. Value of `a+b` will be

Answer

To solve the problem, we need to determine the probability that a nucleus in a sample will decay after four half-lives.

### Step-by-Step Solution:

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

2. **Decay Calculation**: After each half-life, the remaining fraction of undecayed nuclei can be calculated as follows:
- After 1 half-life: \( \frac{1}{2} \) of the original nuclei remain.
- After 2 half-lives: \( \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} \) of the original nuclei remain.
- After 3 half-lives: \( \frac{1}{4} \times \frac{1}{2} = \frac{1}{8} \) of the original nuclei remain.
- After 4 half-lives: \( \frac{1}{8} \times \frac{1}{2} = \frac{1}{16} \) of the original nuclei remain.

3. **Calculating the Probability of Decay**: The probability that a nucleus has decayed after 4 half-lives is given by the fraction of nuclei that have decayed. Since \( \frac{1}{16} \) of the original nuclei remain, the fraction that has decayed is:
\[ 1 - \frac{1}{16} = \frac{16}{16} - \frac{1}{16} = \frac{15}{16} \]

4. **Identifying a and b**: The probability that a nucleus decays after four half-lives can be expressed as \( \frac{15}{16} \). Here, \( a = 15 \) and \( b = 16 \).

5. **Calculating a + b**: Now, we need to find \( a + b \):
\[ a + b = 15 + 16 = 31 \]

### Final Answer:
The value of \( a + b \) is \( 31 \).