# Simple problem in probability

You have 100 lightbulbs. Every lightbulb is either functioning or not. You test 20 of them, and all of the 20 are functioning. What is the probability that 10 of the 100 lightbulbs do not function?

I have confused myself with this problem. I would say that it is impossible to answer, but people I have discussed this problem with all seem to agree on the answer

$\frac{{20 \choose 0}{80 \choose 10}}{{100 \choose 10}}$ or equivalentely, $\frac{{10 \choose 0}{90 \choose 20}}{{100 \choose 20}}$.

This seems weird to me since this would be the way to calculate the probability that none of the 20 lightbulbs you test are nonfunctioning, given that 10 of the 100 are not functioning. Am I only confusing myself?

• You are right, it is impossible to answer without additional hypotheses. – Did Apr 10 '14 at 21:31

Because we have 20 bulbs already functioning those 20 can't be among the 10 which are not functioning. So the 10 non-functioning bulbs will come from the rest (100-20)=80 bulbs. So no of bulbs with this constrain becomes = ${20 \choose 0}$${80 \choose 10}$ . And the total amount is of course ${100 \choose 10}$.Hence this answer.