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**harbottle** Two of the possible outcomes of an experiment are $\displaystyle a$ and $\displaystyle b$, with probabilities $\displaystyle p$ and $\displaystyle q$ respectively and $\displaystyle p+q\leq 1$. If the experiment is repeated until either $\displaystyle a$ or $\displaystyle b$ occurs, what is the probability that $\displaystyle b$ is the first to occur?

My first thought was to work on a convolution of two geometric R.Vs. (ie number of trials since a occur, number of trials since b occurs) but these obviously are not independent since they cannot occur simultaneously.

The solution says: "One can either workk on the joint distribution of the numbers of trials until the first $\displaystyle a$ and $\displaystyle b$ occur, or decompose in terms of the outcome of the first experiment." but I can't work out what it means.

What is the joint distribution? The outcome of the first experiment can be a, b, or something else -- how do I decompose that?

I'm feeling pretty stupid right now. Any help would be appreciated.