White H House of Math logo on multicolored square
Colorful House of Math logo
Log in

What Are Dependent and Independent Events in Math?

Events are independent of each other if one event doesn’t affect the probability of the other event occurring. To find out whether the events A and B are independent of each other or not, you have to check whether the probability of A changes when you already know that B has occurred. This probability is written P (A∣B), and is read as “the probability of A given B”, where the vertical bar between A and B is the sign that means “given”.

Rule

Independence

When A and B are independent events, we have that

P (A∣B) = P (A)

and

P (A ∩ B) = P (A) ⋅P (B)

Rule

Dependence

If A and B are events that depend on each other, we have that

P (A∣B)≠P (A)

and

P (A ∩ B) = P (A∣B) ⋅P (B)

The set A ∩ B in the boxes above is called the intersection of A and B, and the probability P(A ∩ B) is found through the chain rule.

Example 1

Decide whether the two events are independent when you know that

P (smoking) = 0.15, P (lung cancer) = 0.00002, P (lung cancer∣smoking) = 0.00006.

In this case you can simply check whether P (A∣B) = P (A), where A = lung cancer and B = smoking. If it is, the events are independent of each other:

= P (lung cancer∣smoking) = 0.00006 ≠0.00002 = P (lung cancer)

P (lung cancer∣smoking) = 0.00006 ≠0.00002 = P (lung cancer)

Because you got an inequality instead of an equation, you know the events are dependent on each other, which would mean that smoking and getting lung cancer are events that are dependent on each other.

Note! The numbers in this example are made up!

Want to know more?Sign UpIt's free!
White arrow pointing to the LeftPrevious entry
What Are Venn Diagrams in Math?
Next entryWhite arrow pointing to the right
What Is a Conditional Probability?
Go back