The occurrence of the event $A$ is equally likely in every moment of the interval $[0, T]$. The probability of event $A$ occurring at all in this interval is $p$. Given that $A$ hasn't occurred in the interval $[0,t]$ what's the probability that $A$ will occur in $[t, T]$?
I am getting a different answer than the one given in the book. I wonder which one is correct. My answer is
$\dfrac{Tp-tp}{T-pt}$
The book's answer is my answer divided by $p$.
Look at the case when $t = 0$, it is the probability that the event $A$ occurs in $[0, T]$.
You see that the book is mistaken right?
I concur with your answer. It looks like there is a typo in your book.
$\begin{align}\mathsf P(A\in [t,T)\mid A\notin [0,t)) &=\dfrac{\mathsf P(A\notin[0,t)\cap A\in[t,T))}{\mathsf P(A\notin[0,t)\cap A\in[t,T))+\mathsf P(A\notin[0,t)\cap A\notin[t,T))}\\ &=\dfrac{p(T-t)/T}{p(T-t)/T+(1-p)}\\ &=\dfrac{p~(T-t)}{T-pt}\end{align}$
We have $$\Pr(A\in (0,T)\mid A\not\in(0,t))=\frac{\Pr(A\in (t,T))}{\Pr(A\not\in(0,t)}.$$ The top probability is $p\frac{T-t}{T}$ and the bottom is $1-p\frac{t}{T}$; rearranging this gives the same answer you have.
