An amobeba either splits into two or remains the same or eventually dies out immediately after completion of evary second with probabilities, respectively, 1/2, 1/4 and 1/4. Let the initial amoeba be called as mother amoeba and after every second, the amoeba, if it is distinct from the previous one, be called as 2nd, 3rd,...generations.
The probability that after 2 s exactly 4 amoeba are alive is

Options

A. `1//16`
B. `1//8`
C. `3//4`
D. `1//2`

Explanation

To solve the problem, we need to calculate the probability that after 2 seconds, exactly 4 amoebas are alive.

### Step-by-Step Solution:

1. **Understanding the Splitting Process**:
- The amoeba can either:
- Split into 2 amoebas with a probability of \( \frac{1}{2} \)
- Remain the same (not split) with a probability of \( \frac{1}{4} \)
- Die out with a probability of \( \frac{1}{4} \)

2. **Determining Conditions for 4 Amoebas**:
- To have exactly 4 amoebas alive after 2 seconds, the mother amoeba must split into 2 amoebas in the first second, and then each of those 2 amoebas must also split into 2 amoebas in the second second.
- This means that in the first second, we need a split (2 amoebas), and in the second second, both of those must also split (resulting in 4 amoebas).

3. **Calculating the Probability**:
- The probability of the mother amoeba splitting in the first second is \( \frac{1}{2} \).
- The probability of each of the two resulting amoebas splitting in the second second is also \( \frac{1}{2} \) for each.
- Therefore, the total probability can be calculated as follows:
\[ P(\text{4 amoebas alive after 2 seconds}) = P(\text{split at 1st second}) \times P(\text{split at 2nd second}) \times P(\text{split at 2nd second}) \]
\[ = \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{8} \]

4. **Final Result**:
- Hence, the probability that after 2 seconds exactly 4 amoebas are alive is \( \frac{1}{8} \).