I would appreciate help with the following problem, since I can't quite figure out the effect an increasing number of trials has on probability:
Suppose a bin has white marbles and black marbles. Say the probability of choosing a black marble is $P(B) = \beta$. Each experiment consists of taking 5 marbles from this bin. Certainly, the probability that we get no black marbles from one experiment is $(1-\beta)^5$.
Question: If we repeat this experiment of 5 marbles at a time, with replacement, $N$ times then what is the probability that at least one of our $N$ experiments consists of no black marbles (i.e. at least one of our selections is exactly 5 white marbles)? Also, how does this probability grow with $N$?
References, e.g. books or online notes, addressing this theme would also be appreciated!
Best Answer
Let us calculate the probability that none of our selections contain $5$ white marbles. This probability is equal to $\bigl(1-(1-\beta)^5\bigr)^N$ since events are independent. The probability that at least one of our selections is exactly $5$ white marbles is equal to $$ 1-\bigl(1-(1-\beta)^5\bigr)^N. $$