Calculating sample size?

[Andy Resnick]

There has to be a simple way to calculate this, I just don't have a clue how.

The setup:

I'm teaching cell culture methods to some students, and part of the process is checking for contamination- after making culture media, a small volume is put into the incubator overnight and checked for bacterial growth in the morning.

The question (using 'normal' units for things)
"When you made 100 milliliters of solution, there could also be a small number of bacteria present, perhaps as much as 0.1 nanomolar. How much volume do you need to sample in order to be at least 99% confident that you captured at least 1 bacterium?"

Thanks in advance....

 

[jedishrfu]

It seems 1ml would work. I guess you could imagine from 0.1 nanomolar = numerically how many bacteria (6.9E12). Then assume they are evenly distributed.

Take your volume of 100ml and divide it into 100 1ml. So each 1ml would have 1/100th the number of bacteria (6.0E10).

 

[Ibix]

I'd approach it by noting that if there were one bacterium in a volume $V$ ($V=100\mathrm{ml}$ in your case) and I drew a volume $v$, the probability of not getting the bacterium would be $1-v/V$. The probability of not getting any of $n$ bacteria whose locations are independent random variables is therefore $(1-v/V)^n$. Requiring this to be less than 1% gives us $$
\begin{eqnarray*}
0.01&>&(1-v/V)^n\\
0.01^{1/n}&>&1-v/V\\
v&>&V(1-0.01^{1/n})
\end{eqnarray*}$$For your case where the number of bacteria is $n\approx 6×10^{12}$, $v$ is pretty much indistinguishable from zero. You would have to try really really hard not to get a bacterium unless they all clump in one corner or something.

 

[berkeman]

checking for contamination- after making culture media,

Then assume they are evenly distributed.

Is that a valid assumption, though? @Andy Resnick -- what are the typical causes of contamination in this setup? Does it usually come from contaminated dishes (so likely evenly distributed), or contaminated dispensing gear (so likely the contamination will be washed out with the first solution dispensed), or something else?