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Conditional Probability
Common Core: S.CP.3 S.CP.6

Conditional Probability

Author: Sophia Tutorial

This lesson will introduce conditional probability.

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What's Covered

In this tutorial, you're going to learn about conditional probability. This tutorial will cover:

  1. Conditional Probability

1. Conditional Probability

What's the probability of something happening if something else is already happening? Conditional probability is how we figure out that probability.

Term to Know

Conditional Probability

The probability that one event occurs, given that another event has already occurred.

If you are trying to determine the probability that event (B) will occur, given that another event (A) has already occurred. It is written this way:

The probability of B given A. This bar is pronounced, given.

ExampleWhat is the probability of getting an even on a roulette wheel?

This is not a conditional probability yet because the question is simply about the probability that the number is even. To find the answer, count up all of the even sectors.

Notice that zero and double zero don't count as even. The evens are in two categories: even numbers that are in black and even numbers that are in red. 18 of the 38 numbers on the roulette wheel are even.

However, what is the probability that the sector is even, given that the sector is also black? This is a conditional probability statement.

Try It

Ignore any of the sectors that are neither black nor even, and the ones that are only even without being black.

Some of the numbers that are black are also even. So what's the probability of being even?

There are 10 selections that are even and black, out of 18 total black sectors of the roulette wheel. So the probability is: ten even out of 18 total black.

How could we express a formula for the probability of E given B, or the general version with A's and B's in it?

The probability of B given A is equal to their joint probability, A and B, divided by the probability of the thing you're given. The probability that a sector of the roulette wheel is both even and black is 10 sectors out of 38 total sectors, and the probability of being a black sector was 18 out of 38.

In this case, you end up with 10 over 18.


Conditional probability is the probability of some second event occurring, given that some first event has already occurred. It's calculated by dividing the joint probability of the two events by the probability of the existing event, the one that's already happening. This formula works for all events. This isn't a special formula that works only for independent events or only for mutually exclusive events.

Thank you and good luck!


Terms to Know
  • Conditional Probability

    The probability that one event occurs, given that another event has already occurred.