Jumpy, Patient Frog Named Sally And A Math Problem About Her Life-III

Since Sally can only be on $D$ after an odd number of jumps, the probability that Sally is alive after $2k$ seconds is equal to the probability of her being alive after $2k-1$ seconds. Let the probability of this latter be $P_k$ . At this moment she can be located either at $B$ or at $F$ . In two jumps she has a probability of $\frac{1}{4}$ of ending up on $D$ and dying, and therefore a probability of $\frac{3}{4}$ of landing elsewhere and surviving. This means that $P_{k+1}=\frac{3}{4} P_k$ .

$\color{#EC7300}{\text{Induction:}}$

As $P_1=1$ , we can show by induction that $P_k=(\frac{3}{4})^{k-1}$ , where $n=2k-1$ , when $n=2k$ and when $k \geq 1$ .