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I'm trying to figure out the probability of a 3rd failure occurring on the 5th attempt of doing something. Let's just call the probability of success of failure P(S) or P(F), I won't put numbers as I want to actually learn.

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HINTS:

  1. Can you imagine a sequence of successes and failures which would involve the 3rd failure occurring on the 5th attempt. What is the probability of this sequence?
  2. What is the probability of another sequence involving the 3rd failure occurring on the 5th attempt?
  3. How many sequences are there involving the 3rd failure occurring on the 5th attempt? (You could count them as it is certainly in single figures, or you might find an expression)
  4. How do you combine the probabilities of the different sequences?
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    Sequences: 1) FFWWF 2)FWFWF 3) WWFFF 4) WFWFF I don't understand how to combine them2011-03-01
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    @Johann: there is also WWFFF and FWWFF (you seem to have changed from S to W). Can you work out the probabilities of each of these sequences happening in terms of $\Pr (S)$ and $\Pr (F)$?2011-03-01
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    Oh, sorry about the change. Would it be 6p^2(1-p)?2011-03-01
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    No, because $p^2 (1-p)$ is the probability of two successes followed by *one* failure.2011-03-01
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    In that case I'm lost. Could you please explain?2011-03-01
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    You were doing so well. The probability the first is a success is $p$, the probability the second is a success is $p$, the probability the third is a failure is $(1-p)$ so the probability of two successes followed by one failure is $p^2(1-p)$. But the question asks for two successes and *three* failures.2011-03-01
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    So I should multiply them? That would give p^3(1-p), correct?2011-03-01
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    No - you are getting worse, so obviously my advice is not helping and this will be my final comment. $p^3(1-p)$ is the probability of three successes followed by one failure. Your earlier 6 was fine for the six different patterns, but you are having such problems with the probabilities that I think you need to talk to your teacher.2011-03-01
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    Thank you for the help! This definitely means I need some help from my prof. Your patience was great.2011-03-01
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See Negative Binomial Distribution (the section "Alternative formulations").