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This example comes from the book "A First Course in Probability" by Sheldon Ross, Chapter 3, Section 3.5, Example 5C, Page 95. The example is given a solution but there is a part that I don't really understand. I hope someone can explain the logic to me.

The original question is as follow:

Independent trials, each resulting in a success with probability $p$ or a failure with probability $q = (1-p)$, are performed. We are interested in computing the probability that a run of $n$ consecutive successes occurs and before a run of $m$ consecutive failures. We are given the solution as follow:

Solution.: Let E be the event that a run of $n$ consecutive successes occurs before a run of $m$ consecutive failures. To obtain P(E), we start by conditioning on the outcomes of the first trial. That is letting H denote the event that the first trial results in a success, we obtain $$P(E) = pP(E|H)+qP(E|\bar H)$$ Now, given that the first trial was successful, one way we can get a run of $n$ successes before a run of $m$ failures would be to have the next $n-1$ trials all result in successes. So, let us condition on whether or not that occurs. That is, letting F be the event that trials 2 through n all are successes, we obtain $$P(E|H) = P(E|FH)P(F|H)+ P(E|\bar{F}H)P(\bar{F}|H)$$ On the one hand, clearly, P(E|FH) = 1; on the other hand, if the event $\bar{F}H$ occurs, then the first trial would result in a success, but there would be a failure some time during the next $n-1$ trials. However, when this failure occurs, it would wipe out all of the previous successes, and the situation would be exactly as if we started out with a failure. Hence, $$P(E|\bar{F}H) = P(E|\bar{H})$$ Pause !! This is the part where I don't really understand. I agree that when this failure occurs, it would wipe out all of the previous successes. But the situation for E to occur shouldn't be exactly the same as if we started out with a failure because if $F$ is the event that trials $2$ to $n$ are successes, then $\bar{F}$ is the event where not all $2$ to $n$ trials are successes. That being said, the last next $n-1$ trials could be successes. As an example, if the last two of $n-1$ trials is a failure then a success; then the situation for E to occur should be exactly the same as $P(E|H)$. So, can someone explain to me why $P(E|\bar{F}H) = P(E|\bar{H})$ ?

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  • $\begingroup$ Shouldn't $P(\overline{F}H)$ be $P(\overline{F}|H)$ instead, in the second equation? You condition $P(.|H)$ on $F$. Otherwise the whole second product would reduce to $P(E)$. $\endgroup$ May 16, 2017 at 8:02
  • $\begingroup$ @HennoBrandsma Thanks for noticing the error. I have edited the second equation. $\endgroup$
    – M.A.N
    May 16, 2017 at 8:29
  • $\begingroup$ on a side note: if m=n and p=1/2 then the formulas in PDF for P(E) aka P{run of n successes before a run of m failures} and P{run of m failures before a run of n successes} are same, so, they must both give 1/2 (regardless of n) $\endgroup$ Oct 28, 2021 at 20:05

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Since we are looking at an infinite number of trials, after removing a finite prefix of trials, we still have an infinite number of trials. Therefore, we can simply ignore this prefix for the probability of $E$ (even the conditions for $E$ would already be fulfilled by the prefix) and therefore $P(E) = P(E \mid B)$ for any event $B$ that restricts the outcomes of the trials only for a finite prefix.

This hopefully answers your question, but the partial solution you gave above seems rather complicated to me. Thankfully, I was able to find a pdf version of the book via Google. There, the author gives a rather complicated probability for $P(E)$ (not included in your question). That surprised me, because intuitively I thought, $P(E) = 1$.

Thinking about that in more detail, I am now convinced that, indeed, $P(E) = 1$: Let $A$ be the event, that we consecutively have $n$ successes and $m$ failures in $n + m$ trials. Clearly, $P(A) ≠ 0$, because $P(A) = p^n ⋅ q^m$ (assuming $0 < p < 1$).

Now let us iterate these $n + m$ trials and let $E'$ be the event that event $A$ occurs in the first block of $n + m$ trials, or in the second block of $n + m$ trials, or in the third block, etc. Since $P(A) ≠ 0$ and therefore $P(\overline{A}) ≠ 1$ it is easy to see that $P(\overline{E'}) = 0$, since we are looking at an infinite number of trials. Therefore we have $P(E') = 1$.

Since $E' ⊆ E$, it follows that also $P(E) = 1$.

For short: In an infinite sequence of trials you are able to find any finite pattern with probability $1$.

Therefore I suspect that there is a mistake in the solution in the book – unless I made a mistake myself or misunderstood the problem.

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  • $\begingroup$ Would you mind posting the pdf of the book you found? $\endgroup$
    – Jason
    May 16, 2017 at 15:07
  • $\begingroup$ zalsiary.kau.edu.sa/Files/0009120/Files/… $\endgroup$ May 16, 2017 at 15:24
  • $\begingroup$ Thanks, I appreciate it. I've had a quick read and there seems to be numerous errors, and the problem is not very well defined. Your answer seems much better. I'll have a closer read and possibly post my own answer. $\endgroup$
    – Jason
    May 16, 2017 at 16:44
  • $\begingroup$ Regarding your argument for $P(E)=1$: I think you are interpreting the event as "a run of $n$ successes and $m$ failures at some point", whereas the author means "a run of $n$ successes happens before the first run of $m$ failures." $\endgroup$
    – angryavian
    Aug 12, 2023 at 2:11
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I have received a clear explanation and I thought it will be useful to share it out here. $$P(E|\bar{F}H) = P(E|\bar{H})$$ Because, we are not conditioning on the the results of trials 2 through n but rather just on whether they were all successes. That is, we are using

$$P(E|H) = P(E|FH) P(F|H) + P(E|\bar{F} H) P(\bar{F}|H)$$

so in determining $P(E|\bar{F} H)$ we do not know the results of trials 2 through n but only that they were not all heads, that is, that at least one was a tail. Because we know nothing about the sequence following that tail $$P(E|\bar{F} H) = P(E|\bar{H})$$

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@Toni Your answer is correct if the question is asking the probability of $n$ consecutive successes ever occurs before $m$ consecutive failures, i.e. trials do not stop after condition is satisfied. I think the question, may it be phrased ambiguously, was asking if either $n$ consecutive successes of $m$ failures appear, the game stops. If we play such games a large number of times, what's the probability that the game ends with n consecutive successes.

@OP To clarify further, let's draw from the example OP provided where the last two trials of $2^{nd}$ to $n^{th}$ were failure and then success respectively. You argued that if the next $n-1$ trials are success, this is equivalent to $P(E|H)$.

The problem of this argument is that conditioning on $\bar{F}$ does not mean conditioning on the full realization of the $2^{nd}$ to $n^{th}$ trials, but only on if all or not all are successes, i.e. the conditioning doesn't mean the game has proceeded to the point where the $2^{nd}$ to $n^{th}$ trials are all tried. If we ever get a failure within the $2^{nd}$ to $n^{th}$ trials at some point, we are already in the event space defined by $\bar{F}$, then $P(E|\bar{F}H)$ in this case would be conditioning on from that failure. Since whatever happens before that failure is now irrelevant, that's the reason why $P(E|\bar{F}H)$ is exactly the same as $P(E|\bar{H})$.

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