Compound Event In Math Concept & Examples

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Contributors: Mark Lewis, Mia Primas
Author
Mark Lewis
Author:
Mark Lewis

Mark has taught college and university mathematics for over 8 years. He has a PhD in mathematics from Queen's University and previously majored in math and physics at the University of Victoria. He has extensive experience as a private tutor.

Instructor
Instructor:
Mia Primas

Mia has taught math and science and has a Master's Degree in Secondary Teaching.

Learn what a compound event is. Learn about event definition in math. Understand the types of events in math and the compound event definition with examples.

Table of Contents

  • Event in Math
  • Compound Event Definition
  • Probability of Compound Events
Show FAQ

What is simple and compound event?

A simple event is a set of outcomes of a single experiment. A compound event is also the outcome of an experiment, but can be broken down into a combination of simple events occurring simultaneously or in succession.

What is compound event with example?

A compound event is an event which consists of two or more simple events. Drawing five cards from a deck and getting a royal flush is a compound event since it involves the outcome of five individual card draws.

What is a compound event in math?

In math, an event refers to a set of possible outcomes of an experiment. A compound event consists of two of more simple events which occur simultaneously or in succession.

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Event in Math

In mathematics, any repeatable procedure with a fixed set of possible outcomes is called an experiment. The set of all possible outcomes is the sample space of the experiment. An event is by definition any subset of the sample space, meaning an event is a particular set including some outcomes but not others. Events are typically described by some observable condition about the result of the experiment. These ideas of experiments and events are the cornerstones of the field of math known as probability: a common goal is to determine the probability that a particular event has occurred.

Rolling a die, or drawing a card from a shuffled deck, are examples of probabilistic experiments. The sample spaces of each experiment are the set of 6 numbers appearing on the die and the set of 52 different cards in the deck. Any imaginable observation that could be made about the number rolled or card drawn can define an event. Rolling a die may result in the event of rolling a 6, or any even number; drawing a card may produce a face card, or the queen of hearts specifically, to name just a few possible events.

An experiment can also consist of multiple actions or steps: roll the die several times, draw multiple cards, or roll the die while drawing a card simultaneously. To determine the probability that these events or others occur in an experiment, it is essential to distinguish the type of the event as either simple or compound.

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Compound Event Definition

A compound event consists of multiple events which occur simultaneously or in succession. Rolling two dice is a compound event since the outcome of two separate dice determines the outcome. Drawing a five-card hand from a deck of cards is a compound event since it consists of five individual draws made in succession.

Compound events often occur in practical applications since real-life problems are complicated, involving extended chains of events, with multiple effects and contributing causes. For example, doctors and actuaries may be concerned with whether patients have various conditions and co-morbidities, while financial planners may need to prepare for varied economic circumstances.

Simple Events vs. Compound Events

A simple event consists of outcomes of a single experiment. Rolling one die or drawing one card are both examples of simple events. Simple events cannot be reduced into more basic components or steps, while compound events can be considered combinations of two or more simple events.

If the outcomes of an experiment are equally likely, as are rolls of a fair die and draws from a shuffled deck, then the probability of a simple event can be determined by simply counting the number of events where an event {eq}E {/eq} occurs, and dividing by the total number of outcomes:

$$P(E) = \dfrac{ \# \ outcomes \ E \ occurs }{ total \ \# \ outcomes} $$

For example, the probability of rolling a 6 on a single die is {eq}\frac{1}{6} {/eq}, while the probability of rolling an even number is {eq}\frac{3}{6} =\frac{1}{2} {/eq}.

Compound events often consist of so many distinct combinations of outcomes that this formula is impractical for determining the probability. Instead, we can analyze a compound event in terms of its component simple events.

The roll of a die is a simple event, but the roll of multiple dice is compound.

Image of dice.
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Probability of Compound Events

To determine the probability of a compound event, we must distinguish whether it consists of a combination of independent or dependent events.

Events are independent if the occurrence of one event does not affect the probability of the other. Repeated flips of a fair coin are an example of independent events: the odds of a head or a tail remain 50/50 no matter the outcome of the previous flips. Draws from a deck of cards are independent only if the card is returned and the deck is re-shuffled after each draw. If this is not the case, the second draw's outcome depends on which card was removed in the first draw.

The probability of a compound event can be found by multiplying the possibilities of the individual simple events involved. Still, we must take care to distinguish between independent and dependent events.

Hands drawn from a deck of cards are a classic example of dependent events. Each card drawn changes the probability of the next draw.

Image of playing cards.

Compound Probability Formula

If a compound event consists of two independent events {eq}A {/eq} and {eq}B {/eq}, then its probability can be found by multiplying the individual probabilities of each event:

$$P(A , B) = P(A) \times P(B) $$

For example, the probability of flipping two heads on a fair coin is

$$P(H , H) = P(H) \times P(H) = \dfrac{1}{2} \times \dfrac{1}{2} = \dfrac{1}{4} $$

When the two events are independent and consist of relatively few outcomes, it is possible to determine the probability of the compound event using a table. For the previous example, we can tabulate all possible combinations of heads and tails on two coins, using rows and columns to distinguish the two flips:

H T
H HH HT
T TH TT

The table shows that there are four distinct outcomes, precisely 1 of which consists of two heads, for a probability of {eq}1/4 {/eq}.

If one event can influence another then they are dependent. In this case, if one event {eq}A {/eq} occurs, the probability of event {eq}B {/eq} changes to the conditional probability {eq}P(B|A) {/eq}, read as "the probability of {eq}B {/eq} given {eq}A {/eq}". The formula for the probability of the compound event now becomes

$$P(A , B) = P(A) \times P(B|A) $$

We can use this version of the formula to determine the probability of drawing two aces from a deck of cards:

  • There are four aces in the deck of 52, so the probability of drawing the first ace is {eq}P(A) = \frac{4}{52} {/eq}.
  • If that first ace is removed, there are only three aces left among the remaining 51 cards in the deck. The conditional probability of the second ace is {eq}P(A |A ) = \frac{3}{51} {/eq}.
  • According to the formula for dependent events, the probability of drawing the two aces is

$$P(A , A) = P(A) \times P(A|A) = \dfrac{4}{52} \times \dfrac{3}{51} = \dfrac{12}{2652} \approx 0.0045 $$

Instead of a table, a tree diagram can be used to organize information about dependent events. Multiply probabilies along each branch to determine the probability of each possible combination of events A and B.

Tree diagram for dependent events.
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Additional Info

Compound Event Defined

Suppose that your friend Gary misses the bus to class half of the time. We could say that there is a 50% chance that he will miss the bus today. Gary also forgets his homework for 1 out of the 5 classes each week. We know the probability of each even occurring separately, but what if we wanted to find the probability of him missing the bus AND forgetting his homework on the same day?

When we use the term compound event in math, we are referring to the probability of two or more events happening at the same time. In our example, the two events are 1) Gary missing the bus and 2) Gary forgetting his homework. Written mathematically, the probability of this compound event would be:

P(missing the bus, forgetting homework)

Notice that each event is written in parentheses separated by a comma. The P tells us that we are calculating the probability of the events in parentheses.

So how would we find P(missing the bus, forgetting homework)? There are several strategies that can be used to solve problems with compound events. This lesson will show you how to use three common methods.

First, we will use a table to help us see all of the possible outcomes. Next, we will use a tree diagram, which maps out all of the outcomes. Both of these methods are excellent ways to understand the problem in a visual way. The third method involves a formula. Once you are comfortable with the concept of compound event probability, you may prefer to use the formula, which is usually more time efficient.

Finding Probability Using a Table

One way to solve our problem is to set up a table that shows all of the possible outcomes, such as the one shown below.

Using a table to show the possible outcomes
table

Since Gary misses the bus half of the time, we can show this with just two choices, YES and NO. The list across the top represents each of the five days that he goes to class. He forgets his homework one time out of the five days each week, so we represent this by showing YES one time, and NO for the other four times. The boxes in the table are completed to show the possible outcomes of whether Gary misses the bus and forgets his homework.

Out of the 10 possible outcomes, only one of them has a result of YES for both events. This gives us a probability of 1/10 for the compound event.

Finding Probability Using a Tree Diagram

The next strategy involves a tree diagram.

Using a tree diagram to show the possible outcomes
diagram

Each row lists the possibilities of that event happening. For our example, the first row shows the possibility of Gary forgetting his homework for each of the five days. Just as in the table, there is a YES for one day and No for the other four days. Beneath each day, we list the possibilities of him missing the bus for each of those days. There is a 50% chance that he will miss the bus, so we list one YES and one NO to show the possibility of him not missing the bus. Following each line of the diagram shows the different possible outcomes. Just as with the table, there are ten possible outcomes, of which one has yes for both events.

Finding Probability Using a Formula

These first two methods are simple and can help you to visualize the solution. However, it is not practical when you are working with large numbers or several events. For more complex situations, it is useful to use the following formula:

P(A, B) = P(A) x P(B)

Once we know the probability of each individual event, we multiply them together. This gives us the probability of both events happening at the same time. Even if we have more than two events, we just multiply all of their probabilities together.

Let's see how the formula would work for our example.

P(missing the bus, forgetting homework) = P(missing the bus) x P(forgetting homework) = 0.50 x 0.20 = 0.10

Gary has a 10% chance of missing the bus and forgetting his homework on the same day.

Lesson Summary

A compound event in math involves finding the probability of more than one event occurring at the same time. Drawing tables and diagrams are useful strategies when there are only a few events and a small number of outcomes. The formula P(A,B) = P(A) x P(B) can be used for any number of events and outcomes.

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