# Which outcome occurs first?

• Nov 2nd 2010, 07:30 PM
harbottle
Which outcome occurs first?
Two of the possible outcomes of an experiment are $\displaystyle a$ and $\displaystyle b$, with probabilities $\displaystyle p$ and $\displaystyle q$ respectively and $\displaystyle p+q\leq 1$. If the experiment is repeated until either $\displaystyle a$ or $\displaystyle b$ occurs, what is the probability that $\displaystyle b$ is the first to occur?

My first thought was to work on a convolution of two geometric R.Vs. (ie number of trials since a occur, number of trials since b occurs) but these obviously are not independent since they cannot occur simultaneously.

The solution says: "One can either workk on the joint distribution of the numbers of trials until the first $\displaystyle a$ and $\displaystyle b$ occur, or decompose in terms of the outcome of the first experiment." but I can't work out what it means.

What is the joint distribution? The outcome of the first experiment can be a, b, or something else -- how do I decompose that?

I'm feeling pretty stupid right now. Any help would be appreciated.
• Nov 2nd 2010, 07:46 PM
mr fantastic
Quote:

Originally Posted by harbottle
Two of the possible outcomes of an experiment are $\displaystyle a$ and $\displaystyle b$, with probabilities $\displaystyle p$ and $\displaystyle q$ respectively and $\displaystyle p+q\leq 1$. If the experiment is repeated until either $\displaystyle a$ or $\displaystyle b$ occurs, what is the probability that $\displaystyle b$ is the first to occur?

My first thought was to work on a convolution of two geometric R.Vs. (ie number of trials since a occur, number of trials since b occurs) but these obviously are not independent since they cannot occur simultaneously.

The solution says: "One can either workk on the joint distribution of the numbers of trials until the first $\displaystyle a$ and $\displaystyle b$ occur, or decompose in terms of the outcome of the first experiment." but I can't work out what it means.

What is the joint distribution? The outcome of the first experiment can be a, b, or something else -- how do I decompose that?

I'm feeling pretty stupid right now. Any help would be appreciated.

Call the something else outcome C. To get the probability that A appears before B, you need to calculate the sum (using the sum of an infinite geometric series) of the probabilities of the final sequence of events:

A
CA
CCA
CCCA
etc.

I suggest you initially let r = 1 - (p + q) in the calculation.
• Nov 2nd 2010, 08:32 PM
harbottle
Thank you! That makes a lot of sense. The answer would be q/(1-(1-(p+q))) = q/(p+q).