What Is Conditional Probability?

Conditional Probability

Conditional Probability is the probability of event $B$ occurring given that another event $A$ has occurred.

We say this as "the conditional probability of $B$ given $A$", written as $P(B \mid A)$

Sometimes, the concept of conditional probability can be fuzzy to some learners, since there does not necessarily need to be a relationship between $A$ and $B$, nor do they have to occur simultaneously.

Independent Events

If both events occur independently, the conditional probability of event $B$ given event $A$ is just the probability of event $B$ (denoted as $P(B)$) occurring. You will soon learn about using conditional probability to determine independence.

Dependent Events

If events $A$ and $B$ are dependent, then the probability both events occur is written as: $$P(A \thinspace and \thinspace B) = P(A)P(B \mid A)$$

We can rearrange this to get: $$P(B \mid A) = \frac{P(A \thinspace and \thinspace B)}{P(A)}$$

Let's do an example.

Suppose you are waiting to hear from your top college about your acceptance. You have an $85$% chance of being accepted, but dorms are only provided for $75$% of all accepted students. What are the chances you are accepted and will receive a room in a dorm?

$$P(Accepted \thinspace and \thinspace Dorm)= P(Dorm\mid Accepted)P(Accepted)$$ $$= (0.75)\times(0.85) = 0.64$$

Here, we just have to multiply the probability of getting a dorm, given that you are accepted, and also the probability that you are accepted in the first place.

Conditional Probability

We can take our skills of probability to the next level with conditional probability!

In a given sample space, you may encounter a time when one event is dependent on the other. This is when conditional probability comes into play! While these events are not occurring at the same time, they are still related to each other.

Consider a deck of 52 cards. If you select one card and don't replace it in the deck, the probability of the next card you pick will be dependent on the first card that you pick. So: given that you selected a red heart in the your first pick, you can use conditional probability to calculate the probability of selecting a red heart again in your second pick.

We can notate conditional probability as: $$P(B|A)$$

, which we say as: "the probability of event B occurring, given event A has occurred".

####Looking Further

In the next lesson on calculating conditional probability, you will learn how to use Bayes' Theorem and the formula for conditional probability. In the next lesson, you will also learn about two way tables!

We will round out this set of lessons with binomial probability.

(Video) Intro to Conditional Probability

by Trefor Bazett

In this video by Dr. Trefor Bazett, you will learn about the basics of conditional probability.