Compound Events

An event can have more than one step, like flipping a coin and rolling a die. We call these compound events.

To find the probability of a compound event, we first need to figure out how many possible outcomes there are in total.

Listing Possible Outcomes

Example: Coin and Die

Let's say we flip a coin and roll a standard six-sided die:

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

Now we pair every coin flip with every die roll:

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

Count them up ... there are exactly 12 possible outcomes in total.

Listing works beautifully when there are only a few choices, but what if we have lots of them?

The Basic Counting Principle

We can use a clever shortcut to find the number of outcomes using multiplication.

The Basic Counting Principle: If one event can happen in m ways and another event can happen in n ways, the secret is that they can happen together in m × n ways.
Grid showing 3 colorful shirts and 4 pants combined to make 12 distinct outfits.

Example: Mixing and Matching Outfits

Let's try this: imagine we have 3 shirts and 4 pairs of pants in our closet. How many unique outfits can we make?

Instead of drawing every single combination, we just multiply! That means 3 × 4 = 12 different outfits.

OK, we could have counted those, but what about this?

Example: Ice Cream Shop Choices

Wow, look at the menu! We get to choose:

  • 1 of 24 flavors
  • 1 of 7 cones

So what does that mean for our total choices? Let's use our multiplication shortcut:

24 × 7 = 168

There are 168 different ice cream combinations we could order.

Finding Probabilities from Equal Outcomes

So how does counting outcomes help us find probabilities?

Well, if all the outcomes are equally likely to happen, we can use this simple relationship:

Probability = favorable outcomestotal outcomes

Example: Getting a Head and an Even Number

Let's go back to our coin-and-die example. Remember, we already found that there are 12 total outcomes.

What if we want to know the probability of getting a Head and an even number? Even numbers on a die are 2, 4, and 6.

Let's look at our list for our specific "favorable" outcomes:

  • H2, H4, H6

That's 3 favorable outcomes out of the whole bunch.

Probability = 312 = 14

So, the probability is 1 out of 4, or 25%.

Another Compound Event

First, have a play with The Spinner.

Example: Two Spinners

Spinner A has 26 letters of the alphabet, and Spinner B has 10 digits.

We spin both, what's the probability of a letter A to F and an even digit?

To find the total possible outcomes, we multiply the spinner sections together:

26 × 10 = 260

Favorable outcomes are 6 letters A to F, and 5 even digits:

6 × 5 = 30

Which gives us:

Probability = 30260 = 326

Summary

Basic Counting Principle Probability Data Index