Q: Help to understand the thought process of solving the probability trough conditional probability

Question

While studying for my statistics exam i stumbled over the following question:

"There was a warehouse that created bags with marbles in them, they filled all the bags until "enough" marbles was in them. Unfortunately "enough" was not always the same, instead the probability for a bag containing 14, 15 or 16 marbles was the same. Calculate the probability that a bag contains exactly 3 white marbles if the warehouse contains 10% white marbles."

I have no clue how to solve this and tried for a while but just ended up being more confused than i originally was. Is there anyone that can "slowly" explain the thought process of solving a problem like this? I would assume that I need to use some conditional probability but can't really get anywhere with it.

Answer 1

$\Pr(\text{3 White}) = \Pr(\text{3 White}\mid \text{14 in bag})\Pr(\text{14 in bag}) + \Pr(\text{3 White}\mid \text{15 in bag})\Pr(\text{15 in bag})+\Pr(\text{3 White}\mid \text{16 in bag})\Pr(\text{16 in bag})$

Per the law of total probability. If you insist on phrasing it that way, you can also imagine an invisible $\text{"}\mid \text{10% white in warehouse"}$ on every term as well, but it is unnecessary.

Each of these terms in the expansion on the right should be readily able to be calculated.

$\Pr(\text{3 White}\mid\text{14 in bag})$ for instance you can use binomial distribution. $\Pr(\text{14 in bag})$ is given to you in the problem statement.