A Little Math

I was thinking about the this last night.

Suppose you have some situation where you expect (given a lot of repetitions) a result of three, on average.

For instance: Say you had 6 cells and each have a 50% chance of dying over the period of one day. Zero could die, or one etc. up to 6. The average would trend toward three if you did the experiment enough times. There is a way to calculate the probability of getting exactly zero, or one, or whatever number you are interested in.

The formula is here: If you are interested

What concerned me last night was the probability of getting exactly zero deaths and more specifically what would happen if we kept the average at 3 while increasing the sample size: For instance 6 samples with a P=0.5 should be the same as 15 samples with P=0.2 chance of death: The average would be three in either case.

What would be the probability of exactly zero deaths though? Simple. Take the probability of a cell living and raise it to the power of the sample size. This should be obvious--but just in case: Think about flipping a coin. You have p=1/2 of getting heads in one flip. p=1/4 for getting two heads in a row (1/2 x 1/2)

I put the following numbers into Excel:

0.5^6=--------------------0.0156

0.7^10=-------------------0.0282

0.8^15=-------------------0.0351

0.9^30=-------------------0.0423

0.99^300=-----------------0.0490

0.999^3,000=--------------0.04971

0.9999999^30,000,000=---0.04978

Please note that I use the P for living since for there to be zero deaths, all must live. If the P for death = 0.2 the P for life = 0.8. You either live or die and the total must be 1.0 or 100%.

I am not sure why the results look like this: The probability of exactly zero deaths goes from 1.5% at the lowest end to approaching 5% at the high end, Why? (Bear in mind too, for any of these sample size probability sets, the average will always be three cell deaths). I would have thought at the low end it would be more likely to get zero since with an N of 6 there are only seven possible results: 0, 1, 2...6. At the high end there are 30 million possible results--even if some of them are pretty remote.