To calculate the probability of event $B$ given that event $A$ has occurred, we use the formula:

Conditional probability allows us to understand the likelihood of an event occurring based on the occurrence of another event. In this context, we are interested in determining the probability of event $B$ occurring, given that event $A$ has already taken place. The components of the formula are as follows:

• $P(B|A)$: the conditional probability of event $B$ given event $A$.
• $P(A \text{ and } B)$: the probability that both events $A$ and $B$ occur simultaneously.
• $P(A)$: the probability that event $A$ occurs.

Let’s illustrate this with a practical example. Consider a standard deck of 52 playing cards, and we want to find the probability of drawing a King (event $B$) given that we have already drawn a red card (event $A$).

1. First, we calculate $P(A)$, the probability of drawing a red card. There are 26 red cards in a deck of 52, so:

2. Next, we determine $P(A \text{ and } B)$, which is the probability of drawing a red King. There are 2 red Kings in the deck, thus:

3. Now, we can apply the conditional probability formula:

Therefore, the probability of drawing a King given that we have already drawn a red card is $\frac{1}{13}$. This approach can be applied to any scenario where you need to find the conditional probability of one event based on another event.