Compound Events

Sometimes an event has more than one step. These are called compound events.

To find probabilities for compound events, we first need to know how many possible outcomes there are.

Listing Possible Outcomes

Let's start with a simple example.

Example: Coin and Die

We flip a coin and roll a six-sided die.

  • The coin can land on: Heads (H) or Tails (T)
  • The die can land on: 1, 2, 3, 4, 5, 6

We can list all the possible outcomes:

  • H1, H2, H3, H4, H5, H6
  • T1, T2, T3, T4, T5, T6

There are 12 possible outcomes in total.

Listing outcomes works well when there are only a few choices.

The Basic Counting Principle

When listing becomes too long, we can use a quicker method.

Basic Counting Principle: If one event can happen in m ways and another event can happen in n ways, then together they can happen in m × n ways.
3 shirts x 4 pants = 12 choices

Example: you have 3 shirts and 4 pants.

That means 3 × 4 = 12 different outfits.

Example: Ice Cream Choices

You choose:

  • 1 of 3 flavors
  • 1 of 2 cones

Number of possible outcomes:

3 × 2 = 6

There are 6 different ice cream combinations.

Finding Probabilities from Equal Outcomes

If all outcomes are equally likely, we can find probability using:

Probability = favorable outcomes ÷ total outcomes

Example: Getting a Head and an Even Number

From our coin-and-die example, there were 12 total outcomes.

Even numbers on the die are: 2, 4, 6

Favorable outcomes:

  • H2, H4, H6

That's 3 favorable outcomes.

Probability = 312 = 14

So, the probability is 1 out of 4.

Another Compound Event

Example: Spinners

Spinner A has 4 colors.
Spinner B has 3 numbers.

Total possible outcomes:

4 × 3 = 12

If only 2 outcomes are winning outcomes, then:

Probability = 212 = 16

Summary

Basic Counting Principle Probability Data Index