Thursday, April 10, 2008

Probability Confuses Me

My boss asked me the question "If a track has a 1/15 probability of playing, what is the probability that it will play in a sequence of 15 tracks" (yes, this has to do with on-line radio). The simple answer is the basic binomial case, or what is the probability of 1 success in 15 tries given that the probability of success is 1/15, or (15!)/1!14!*(1/15)^1*(14/15)^14. That works out to about a 40% probability.

However, the answer desired could be the probability that the track plays 1 or more times, which is closer to 65%. However, in the specific case of radio the rules say that a track can't play more than once in a certain span, so we're back to the original case if the separation is large enough. On the other hand, the binomial case assumes that the probability of a track playing stays the same, but if the track is chosen randomly from a fixed pool, and that pool become progressively smaller as tracks are chosen, then the probability changes. For example, in the case where you've only got 15 tracks and 15 spots to put them in, then the probability that the track will play assuming repeats aren't allowed is 1.

This is why probability confuses me: many times figuring out the question is harder than finding the answer.

2 comments:

William said...

True.. Probability is an enigmatic subject. The best way to answer that problem is to observe that the prob of at least one success is the same as one minus the prob of no successes. For that problem, that comes out to 1 - (14/15)^15 ≈ 64%.

William said...

Oh I see. I am sleepy and did not read that you also got what I said. :)

As far the second scenario goes... maybe you could view it as a poisson arrival process, with the time distributed according to a minimum interval between repeats plus a poisson term. What caused you to think about this?