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Let A and B be two mutually exclusive events with positive probability associated with some random experiment. The experiment is replicated until event B is observed. Find the probability that by that time event A was already observed twice.

I'm not sure where to begin with this problem, any insights?

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  • $\begingroup$ Are the event A and B the only outcomes of the experiment? $\endgroup$
    – Lei Hao
    Dec 6, 2016 at 4:52
  • $\begingroup$ No. They are just two events with positive probability. $\endgroup$
    – ViC
    Dec 6, 2016 at 4:54
  • $\begingroup$ Exactly twice or at least twice? $\endgroup$
    – copper.hat
    Dec 6, 2016 at 5:45
  • $\begingroup$ Exactly twice.. $\endgroup$
    – ViC
    Dec 6, 2016 at 5:46

2 Answers 2

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Since we aren't interested whether a relative sequence ...A...A...B occurs on the third trial or the hundreth, the events other than $A$ and $B$ are irrelevant, and we can greatly simplify the problem.

Let the odds in favor of $A$ over $B$ be $3:2$, say,
then we can take $p = 0.6, q = 0.4$,
and the required probability is $0.6^2\cdot0.4$


Added

As an example,suppose we repeatedly roll a die,
let A = getting a number divisible by $3$,
let B = getting $5$,

then odds in favor of A $=2:1$,
$p=2/3, q = 1/3$,
Pr ...A...A...B $= \left(\dfrac 23\right)^2\cdot\dfrac13 = \dfrac2{27}$

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  • $\begingroup$ To clarify. $~$ We want the probability that, of all events where either $A$ or $B$ occurs, the first $B$ will occur on the third such. $$\mathsf P(A\mid A\cup B)^2\,\mathsf P(B\mid A\cup B) =\dfrac{\mathsf P(A)^2\,\mathsf P(B)}{(\mathsf P(A)+\mathsf P(B))^3}$$ $\endgroup$ Dec 6, 2016 at 8:54
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Suppose event B occurs at time $k$ with probability $q$, event A's occurrence before $k$ is a binomial distribution ~ Binomial$(k-1,p)$. So the probability is $$P(\text{A occurs twice before B} )=\sum_{k=3}^\infty {k-1\choose2}p^2q(1-p-q)^{k-3}$$

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