Exponential back-off algorithm is in use here. Here, when a collision occurs:
- each sender sends a jam signal and waits (back-offs) $k \times 51.2 \mu s$, where $51.2$ is the fixed slot time and $k$ is chosen randomly from $0$ to $2^N-1$ where $N$ is the current transmission number and $1\leq N\le 10.$ For $11\leq N \leq 15, k$ is chosen randomly from $0$ to $1023.$ For $N=16,$ the sender gives up.
For this question $N=1$ for $A$ as this is $A's$ first re-transmission and $N = 2$ for $B$ as this is its second re-transmission. So, possible values of $k$ for $A$ are $\{0,1\}$ and that for $B$ are $\{0,1,2,3\},$ giving $8$ possible combinations of values. Here, $A$ wins the back-off if its $k$ value is lower than that of $B$ as this directly corresponds to the waiting time. This happens for the $k$ value pairs $\{(0,1), (0,2), (0,3), (1,2), (1,3)\}$ which is $5$ out of $8$. So, probability of $A$ winning the second back-off race is $\frac{5}{8} = 0.625.$
Correct Answer: $B$