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 amoeba population will be maximum after completion of 3 s is

Options

A. `1//2^(7)`
B. `1//2^(6)`
C. `1//2^(8)`
D. None of these

Answer

A

Explanation

If first second, mother amoeba splits into two with the probability 1/2.
In 2nd second, tow amoeba split into two with probability `(1//2)xx(1//2)=1//4` In 3rd second, four amoeba split into two with probability
`(1//2)xx(1//2)xx(1//2)xx(1//2)=1//16`
Hence, the probability that the population is maximum after 3 s is `(1//2)xx(1//4)xx(1//6)=1//2^(7).`