Simple, compound, and complementary events are different types of probabilities. Each of these probabilities are calculated in a slightly different fashion. In this lesson, we will look at some real world examples of these different forms of probability.

## Probability of Simple Events

The **probability of simple events** is finding the probability of a single event occurring.

When finding the probability of an event occurring, we will use the formula: number of favorable outcomes over the number of total outcomes. After calculating this probability, your answer will need to be in simplest form.

## Example of Simple Events

Let’s look at an example of simple events. Sam owns a large fish store with many colors of fish. He keeps all of the fish in a large aquarium. In his main aquarium, he has 5 red fish, 6 blue fish, 14 white fish and 5 green fish. A customer comes into the store and wants to buy a blue fish to take home.

What is the probability that Sam will reach in and scoop out a blue fish on his first scoop?To calculate this probability, Sam needs to find out the total number of fish in his aquarium. Sam needs to add the 5 red fish + 6 blue fish + 14 white fish + 5 green fish = 30 total fish. Sam must also know the number of favorable outcomes. In his tank, there are only six blue fish. So his number of favorable outcomes is six.

The formula to calculate the probability is the number of favorable outcomes over the number of total outcomes. The number of favorable outcomes is six and the number of total outcomes is 30, so, the probability of Sam scooping a blue fish the first time is 6 out of 30. Remember, all fractions must be in simplest form, so 6 over 30 will reduce to 1 over 5. The probability that Sam will scoop out a blue fish on his first try is 1/5.

## Probability of Compound Events

**Compound events** are a bit more complex than the simple events in the last example. Compound events involve the probability of more than one event happening together.

With compound events, we will use the same formula to calculate the probability of each event occurring. To calculate the probability, we will use the formula: number of favorable outcomes over the number of total outcomes. Once we find the probability of each event occurring, we will multiply these probabilities together.

## Example of Compound Events

Let’s look at an example of compound events to see how to calculate more than one event occurring.

Wendy and Kim are playing a new deluxe board game titled ED Portalopoly. It is Wendy’s turn and she needs to roll the two dice and get both to be a six in order to land on the jackpot. What is the probability that Wendy will roll a six and then roll another six?To start this problem, we need to calculate each event separately. The total number of outcomes on the dice is six and the number of favorable outcomes is one, because there is only one six on a dice. So, the probability of Wendy getting a six on her first roll is 1 out of 6.

Next, we need to calculate the probability of Wendy getting a six on her second roll. Since these dice are the same, the probability of getting a six on the second roll will also be 1 out of 6. Now that we know the probability of both events happening, we need to multiply these two fractions together. So 1/6 x 1/6 = 1/36. So, the probability of Wendy rolling both dice and getting a six on both is 1/36.

## Probability of Complementary Events

**Complementary events** are another type of event in which we can calculate the probability.

Complementary events are events that add together to equal a whole or one. For example, if the probability of it raining today were 2/5, what would the probability be of it not raining? The probability of it not raining would be 3/5, because 2/5 + 3/5 would equal 5/5 or one whole. We know that there are only two choices: it will either rain or not rain.

## Example of Complementary Events

Let’s check back on Wendy and Kim playing everyone’s favorite board game, ED Portalopoly. It is now Kim’s turn to roll.

If she rolls two dice that add together to equal eight, she will land in jail. Wendy is hoping that Kim lands in jail on this turn. What is the probability that Kim will not roll two dice whose sum is eight?Wendy needs to first calculate the probability that Kim will roll the two dice and get a sum of eight. To find this probability, she will use the formula: number of favorable outcomes over the number of total outcomes.Since Kim is rolling two dice and each dice has six sides, there are 36 total outcomes that she can get.

To find the number of favorable outcomes, Wendy decides to list out possible ways that the two dice can add to eight. She knows that Kim could roll a 2 + 6, or a 3 + 5, or a 4 + 4, or a 5 + 3 or a 6 + 2. She can see that there are only five different ways to roll two dice whose sum will be eight. Wendy now knows that the probability of Kim rolling two dice and getting a sum of eight is 5 out of 36, or 5/36.Wendy also wants to know the probability of Kim not getting the two dice to add together to equal eight.

These two events are complementary, because they will equal 1 or 36/36. Since the probability of Kim getting the sum of eight was 5/36, we can subtract to find the probability of her not getting the sum of eight.36/36 – 5/36 = 31/36. The probability of Kim not rolling the two dice and getting the sum of eight is 31/36, so the chance that Kim does not land in jail on her next turn is 31/36.

## Lesson Summary

In this lesson, we looked at three different types of probability. The probability of **simple events** is finding the probability of a single event occurring. When finding the probability of an event occurring, we will use the formula: number of favorable outcomes over the number of total outcomes.**Compound events** involve the probability of more than one event happening together.

With compound events, we will use the same formula to calculate the probability of each event occurring. Once we find the probability of each event, we will multiply these probabilities together. And the last one was complementary events. **Complementary events** are events that add together to equal a whole or one.

## Learning Outcomes

After finishing this lesson, you should be confident in your ability to solve for probability in simple, compound and complementary events.