Conditional Probability Study Guide

Conditional probability is the probability of an event occurring given that another event has already occurred. It is denoted by P(A|B), which reads as "the probability of event A given event B."

Formula

The formula for conditional probability is:

P(A|B) = P(A and B) / P(B)

Where:

P(A|B) is the conditional probability of A given B
P(A and B) is the probability of both A and B occurring
P(B) is the probability of event B occurring

Example

Suppose you have a bag with 3 red balls and 2 green balls. If you pick a ball at random, the probability of picking a red ball is 3/5 (since there are 3 red balls out of 5 total balls).

Now, let's say you pick a red ball from the bag. If you were to pick another ball, the probability of picking a red ball again would be 2/4, as there are now 2 red balls left out of 4 total balls.

This is an example of conditional probability - the probability of picking a red ball on the second draw given that a red ball was picked on the first draw.

Practice Problems

What is the probability of drawing a king from a standard deck of cards, given that the card drawn is a heart?
Suppose a jar contains 5 blue marbles and 4 red marbles. If you draw a marble, what is the probability of drawing a blue marble, given that the first marble drawn was red and not replaced?

Study Tips

To master conditional probability, it's important to understand the concept of independence and dependence of events. Additionally, practice solving different types of conditional probability problems to strengthen your understanding of the topic.