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!
 A: 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
$$
1-\bigl(1-(1-\beta)^5\bigr)^N.
$$
A: Increasing the number of trials will impact probability if you are sampling without replacement. This makes intuitive sense. If I do not replace after I draw a sample, the probability increases since the fraction of sample over population increases (population went down).  If I sampled with replacement the population would stay the same, thus the probability would remain unaffected. 
For example, if my name is in a hat with 6 other names, I have a 1/7 chance of drawing my name. If I replace my name into the hat, then pass the hat to you, you also have a 1/7 chance of drawing your name. No increase in probability. However, should I not replace my name after drawing it from the hat, then pass the hat to you, you would now have a 1/6 chance of drawing your name. Sampling without replacement increases the probably. 
