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I am reading a probability text book and I am having difficulty understanding a paragraph and how the equation was setup. Can anyone please explain it in more detail or show me how this was derived? Please see the hyperlink to the screenshot.

Screenshot

"Equation 2-3" is the conditional probability P[A|B] = P[AB]/P[B]

What I think I understand:

  • I understand the formula P[A] = pa + qB
  • The first trial is a success, so I understand p^(s-1) term

I lose understanding at the Bth trial and anything after that, its probability q*p^(B-2), and conditional probability of A. The highlighted parts are what I am having trouble with.

Any help or clear way to understand will be appreciated very much.

Thank you

Textbook: The Theory of Gambling and Statistical Logic by Richard A. Epstein

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1 Answer 1

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I think it is either bad typography or bad choice of variables to choose the same variable name for both probability and number of first failure.

So, let's see what can happen if the first trial was success:

  1. Next $s - 1$ trials are also successes, probability of it happening is $p^{s - 1}$ and probability of win in this case is $1$.
  2. Second trial results in failure. Probability of it is $q$ (remember, we condition on first trial been success), and probability of win in this case is $\beta$ (we now have the same situation as we would have if the first trial failed).
  3. Third trial results in failure. Probability of it is $p\cdot q$, and probability of win in this case is $\beta$. ... s. $s$-th trial results in failure. Probability of it is $p^{s - 2}\cdot q$, and probability of win in this case is again $\beta$.

This cases are mutually exclusive, so to get probability of win given the first trial was success we need to sum them: $p^{s - 1} + q\beta + pq\beta + \ldots + p^{s - 2}q \beta$.

Although it would probably be simpler to just note that after first success with probability $p^{s-1}$ we win on $s$-th trial, and with probability $1 - p^{s - 1}$ we fail one of first $s$ trials, and after this probability of winning is $\beta$.

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