This question has been asked before. Here is the link: Mutually exclusive events

Here is the description to the problem:

Let E and F be mutually exclusive events in the sample space of an experiment. Suppose that the experiment is repeated until either event E or event F occurs. What does the sample space of this new super experiment look like? Show that the probability that event E occurs before event F is P(E)/ [P(E) + P(F)]. Hint: Argue that the probability that the original experiment is performed n times and E appears on the nth time is P(E)×(1−p)n−1, n = 1, 2, . . . , where p = P(E) + P(F). Add these probabilities to get the desired answer.

It appears to me that this is an example of a "Geometric Distribution". I am trying to solve this using the hint given, but I know not how to proceed, especially with adding up the probabilities as suggested.

Thank you,


1 Answer 1


Assuming that at least one of $E$ or $F$ has positive probability, then with probability $1$, one of them will occur eventually.

Then $P(E|(E\cup F))=\frac{P(E\cap(E\cup F))}{P(E\cup F)}=\frac{P(E)}{P(E)+P(F)}$

  • $\begingroup$ Could you please explain your choice of answer? Why did you pick up $P(E|E \cup F)$ $\endgroup$
    – Quester
    Jan 28, 2015 at 5:11
  • 1
    $\begingroup$ @Quester: Eventually $E$ or $F$ will occur (that's what we're waiting for in the process), and that event is $E\cup F$. One way to think of it would be if someone else was doing the experiment, and after each trial that person told us: 'yes, $E$ or $F$ occurred'; or 'no, neither $E$ nor $F$ occurred'. Eventually the person would say 'yes', and we want to know the probability of that the eventual 'yes' would come from $E$ occurring, rather than from $F$ occurring. $\endgroup$
    – paw88789
    Jan 28, 2015 at 11:35
  • $\begingroup$ @paw..Makes sense thank you $\endgroup$
    – Quester
    Jan 29, 2015 at 0:23

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