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

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

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.