A particular nucleus in a large population of identical radioactive nuclei did survive 5 halt lives of that isotope. Then the probability that this surviving nucleus will service the next half life is

Options

A. `(1)/(32)`
B. `(1)/(5)`
C. `(1)/(2)`
D. `(1)/(10)`

Answer

To solve the problem, we need to determine the probability that a radioactive nucleus, which has already survived for five half-lives, will survive the next half-life.

### Step-by-Step Solution:

1. **Understanding Half-Life**:
- The half-life of a radioactive isotope is the time required for half of the radioactive nuclei in a sample to decay. For any given nucleus, the probability of surviving one half-life is 50% (or 0.5), and the probability of decaying (not surviving) is also 50% (or 0.5).

2. **Survival of the Nucleus**:
- The problem states that the nucleus has survived for five half-lives. This means that it has not decayed during each of those five intervals. However, the survival of the nucleus in previous half-lives does not affect its probability of surviving the next half-life.

3. **Probability Calculation**:
- Since the decay of a radioactive nucleus is a random process, the probability of survival for the next half-life remains constant at 50%, regardless of how many half-lives it has already survived. Therefore, the probability that the nucleus will survive the next half-life is still 0.5.

4. **Conclusion**:
- The probability that the surviving nucleus will survive the next half-life is **0.5** or **50%**.

### Final Answer:
The probability that the surviving nucleus will survive the next half-life is **0.5**.

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