# What is the probability of two events occuring in time $T$ if $b$ cannot occur until $a$ does?

There are two events with equal or distinct probabilities of occurring in time $T$, $P_{a}(t)$ and $P_{b}(t)$. Event $b$ cannot occur until event $a$ occurs. How do you calculate $P_{ab}(T)$ where $T$ is the total time in which both events must occur in?

To simplify matters, assume that $P_{a}(t) = P_{b}(t) = 0.25$ and $T = 5$. Starting from $t=0$ and incrementing $t$ by $1$ until $t=T=5$ what ends up happening is that $P_{b}(t)=P_{a}(T-t)$ resulting in a parabolic graph. (The reason for this is that the sooner $a$ occurs, the more time $b$ has to occur). Calculating the chances of both occurring at exactly $t=$ some number is trivial, but what is the overall probability of both occurring over a period of $T$?

There are several operations I know I could use, but I'm unsure as to which one would be correct. Do I need to XOR all the values for every increment in time? Do I take a mean or median of those values?

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