# Repeated sampling with replacement, increasing probability

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!

• Welcome at math.SE! It might make things easier if you summarize your five-picks-experiment as one process with sucess probability $\beta'=(1-\beta)^5$. Mar 6, 2014 at 13:51

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 $$1-\bigl(1-(1-\beta)^5\bigr)^N.$$
• thank you! with $\alpha = P(W)$, probability of white, so $\alpha+\beta= 1$, then $P$(no selection of $N$ experiments has exactly 5 white) $=1-(1-\alpha^5)^N$. This leads me to your final answer. thanks again Mar 6, 2014 at 13:58
• does one say the "growth" of $1-(1-\alpha^m)^N$, for $m$ fixed, is "exponential in $N$"? Mar 6, 2014 at 14:00
• @fractalEd You're welcome! I would say that it converges to $1$ exponentially fast if $0<\beta<1$. Mar 6, 2014 at 14:17