Need some math help (odds of success)

Nich

I've been banging my head against this for a while, and I feel like it should be really easy, but I'm struggling to figure out how to do it.

I need a formula which describes the number of ways to succeed at a task, given a number of tries.

For example, flipping 3 coins and getting at least two heads, or flipping 4 coins and getting at least two heads

For the first one, enumerating the options gives 4 ways of succeeding and 4 ways of failing, and for the second I count 11 ways to succeed and 5 ways to fail, but I can't seem to find a formula that describes how many ways to get m heads in n tries.

Pi-r8

What you are looking for is described by the binomial distribution. If it makes you feel better, it's a pretty hard problem.

Bliss 101

You can do this easily in OpenOffice Calc (and Excel too, but I don't have excel at hand right now). The function you want is B().

Syntax:
=B(trials, probability_success, min_successes, <max_successes>)

If you omit the max_successes parameter, you get the probability of succeeding exactly min_successes times. Otherwise you get the probability to succeed between min_successes and max_successes times. So if max_successes equals trials, you're calculating the probability of succeeding at least min_successes times.

So for your second example B(4, 0.5, 2, 4) gives the probability 0.6875 for getting at least two heads. If the probability to succeed at any given trial is exactly 0.5, you can then use this to calculate the number of combinations: your total number of tests is trials^2, or 16 in this case, so just multiply that by the probability from the beta distribution: 0.6875 * 16 = 11 possible ways to succeed.

If your probability of success at an individual trial is something other than 0.5, then the proportion of possible winning combinations and the probability of overall success are no longer the same. But if you really need the number of possible combinations, you can still calculate the number of winning combinations by using 0.5 as your success rate.

edit: in Excel the function to use would be BINOMDIST(). The syntax is slightly different, and it calculates the cumulative distribution (at most N successes), so to get the probability for at least N successes you need to use 1-BINOMDIST.

Nich

Ah, yes, perfect, thanks. Googling binomial distribution gets exactly what I'm looking for.
I'm 99% sure I studied this in statistics, but like most of my math classes I promptly forgot everything after the exam.