Question

What does p(b|a) represent? the probability of event a or event b or both occurring. the probability of event a and event b both occurring. the probability of event b occurring after it is assumed that event a has already occurred. the probability of event a occurring after it is assumed that event b has already occurred?

Answer 1

$P(B|A)$ represents the probability of the event B, knowing that event A has already occurred. It is called conditional probability.

Conditional probability can heavily change the probability of an event, even if you know its apriori probability. Consider this example.

We're flipping two coins, and we define the following events:

$A = \text{The first coin lands on tails},\quad B=\text{The first coin lands on heads},\quad C = \text{Both coins land on heads}$

Before flipping the two coins, we know that

$P(C) = \dfrac{1}{4}$

In fact, there are four outcomes with the same probability: $\{HH, HT, TH, TT\}$

But what if we condition the probability of C to events A or B? We have

$P(C|A) = 0$

In fact, there's no way that both coins can land on heads, if the first one landed on tails. On the other hand,

$P(C|B) = \frac{1}{2}$

Because now that we know that the first coin landed on heads, there are only two possible outcomes: $\{HH, HT\}$