Probability of getting SUCCESS AND FAILURE at number n-1 and n trial

Question

In a sequence of Bernoulli trials let $u_n$ be the probability that the combination SF occurs for the first time at the trials number n-1 and n. To find the generating function I wrote the following recurrence relation but need verification. $$u_n=(1-u_{n-2})pq$$ I would much appreciate if someone could help me derive the generating function and verify the above recurrence relation.Thanks in advance

Answer 1

The combination SF appears for the first time at times $n-1$ and $n$ if and only if two conditions are met:

• Trials $n-1$ and $n$ are S and F.
• Trials from $1$ to $n-2$ are a sequence of $i$ trials F followed by $n-2-i$ trials S, for some $0\leqslant i\leqslant n-2$.

Hence, $$ u_n=\sum_{i=0}^{n-2}q^ip^{n-i-1}q=qp^{n-1}\sum_{i=0}^{n-2}\left(\frac{q}p\right)^i=\ldots$$ Edit: A recursive formula which can be used to compute the sequence $(u_n)_{n\geqslant1}$ is that $u_1=0$ and, for every $n\geqslant0$, $$ u_{n+2}=pq(1-U_{n}),\qquad U_n=\sum_{i=1}^nu_i. $$