Bacterial Sampling : Dependent event

Question

I have a question regarding distribution of bacteria in wells: I have a tube that contains 200ul of 5000CFU/ml of bacterial solution.

If I have 20 tubes to which I will add 10ul of bacterial sample from the 200ul original tube, what is the probability that the conc of bacteria in the well is between 30-60? I reason I have problem with this is because, the 10ul taken out is not put back in there, so this should be considered a dependent event, so I can't use the Poisson's distribution, right? How to calculate this probability then?

Answer 1

You apparently have $1000$ bacteria in your big tube. Is there any variation in this? If there is then that would push the variance up slightly. Let's assume not.

You could model the number of bacteria in a little tube as a binomial distribution with parameters $1000$ and $\frac{1}{20}$ so with mean $50$ and variance $47.5$. The probability of having from $30$ through to $60$ inclusive, using R, would be

> pbinom(60, 1000, 1/20) - pbinom(29, 1000, 1/20)
[1] 0.9322183

while using a Poisson approximation with the same mean (implicitly a slightly higher variance of $50$, slightly reducing the probability of being in a central interval) would give

> ppois(60, 50) - ppois(29, 50)
[1] 0.926923

or using a Gaussian approximation with a continuity correction and the binomial mean and variance would give

> pnorm((60.5 - 50)/sqrt(47.5)) - pnorm((29.5 - 50)/sqrt(47.5)) 
[1] 0.9347156

all giving a probability of about $93\%$.