Probability Rules

Basic Probability Rules

In the previous section we considered situations in which all the possible outcomes of a random experiment are equally likely, and learned a simple way to find the probability of any event in this special case. We are now moving on to learn how to find the probability of events in the general case (when the possible outcomes are not necessarily equally likely), using five basic probability rules. Fortunately, these basic rules of probability are very intuitive, and as long as they are applied systematically, they will let us solve more complicated problems; in particular, those problems for which our intuition might be inadequate.

Rule 1

For any event A, 0 ≤ P(A) ≤ 1.

This first rule simply reminds us of the basic property of probability that we’ve already learned. The probability of an event, which informs us of the likelihood of it occurring, can range anywhere from 0 (indicating that the event will never occur) to 1 (indicating that the event is certain). One practical use of this rule is that is can be used to identify any probability calculation that comes out to be more than 1 as wrong.

Rule 2

P(S) = 1; that is, the sum of the probabilities of all possible outcomes is 1.

Complement Rule

Let’s move on to rule 3. In probability and in its applications, we are frequently interested in finding out the probability that a certain event will not occur. An important point to understand here is that “event A does not occur” is a separate event that consists of all the outcomes in the sample space S that are not in A. It is for this reason that the event “event A does not occur” is called “the complement event of A,” since it compares event A to the whole sample space. Notation: we will write “not A” to denote the event that A does not occur. Here is a visual representation of how event A and its complement event “not A” together represent the whole sample space.

The entire sample space S is represented with a gray box. Inside of this box is a blue circle, representing all outcomes in A. Everything else in the gray box but outside of the blue circle is "not A".

Comment

Such a visual display is called a “Venn diagram.” A Venn diagram is a simple way to visualize events and the relationships between them using rectangles and circles. We will use Venn diagrams throughout this module.

Rule 3 deals with the relationship between the probability of an event and the probability of its complement event. Given that event A and event “not A” together make up the whole sample space S, and since rule 2 tells us that P(S) = 1, the following rule should be quite intuitive:

Rule 3: The Complement Rule

P(not A) = 1 – P(A); that is, the probability that an event does not occur is 1 minus the probability that it does occur.

We are now moving to rule 4, which deals with another situation of frequent interest, finding P(A or B), the probability of one event or another occurring. Before we get to the actual rule, however, we need some clarifications and definitions.

When a parent says to his or her child in a toy store “Do you want toy A or toy B?”, this means that the child is going to get only one toy and he or she has to choose between them. Getting both toys is usually not an option.

In contrast,

In probability, “OR” means either one or the other or both.

and so,

P(A or B) = P(event A occurs or event B occurs or both occur)

Having said that, it should be noted that there are some cases where it is simply impossible for the two events to both occur at the same time, in which case we don’t have to worry about the possibility that both occur when we try to find P(A or B). The distinction between events that can happen together and those that cannot is an important one.

Consider the following two events:

A—a randomly chosen person has blood type A

B—a randomly chosen person is a woman.

In this case, it is possible for events A and B to occur together.

Definition: Two events that cannot occur at the same time are called disjoint or mutually exclusive. (We will use disjoint.)

We can therefore say that in the first example events A and B are disjoint, and in the second example they are not disjoint. Using Venn diagrams, we can visualize two events that are disjoint and compare them to two events that are not:

A Venn diagram titled "A and B are Disjoint." The entire sample space is represented as a rectangle. Inside the rectangle are two separate circles. One circle represents the events in A and the other represents the events in B.
A Venn diagram titled "A and B are NOT Disjoint." The entire sample space is represented as a rectangle. Inside the rectangle are two circles. One circle represents the occurrences in A and the other represents the occurrences in B. These two are not disjoint, so the two circles partially overlap each other. (Being NOT disjoint, two circles could overlap each other completely, but in this example they do not.)

The Venn diagrams suggest that another way to think about disjoint versus not disjoint events is that disjoint events do not overlap. They do not share any of the possible outcomes, and therefore cannot happen together. On the other hand, events that are not disjoint are overlapping in the sense that they share some of the possible outcomes and therefore can occur at the same time.

The purpose of the following activity is to strengthen your intuition and understanding about disjoint versus not disjoint events.

Addition Rule for Disjoint Events

Now that we understand the idea of disjoint events, we can finally get to rule 4. Rule 4 actually has two versions, one for finding P(A or B) in the special case when events A and B are disjoint, and a more general version for when the events are not necessarily disjoint. We will first present the version of rule 4 that is restricted to disjoint events, and later in the section (after rule 5) we will revisit rule 4 and present the more general version.

Rule 4: The Addition Rule for Disjoint Events

The Addition Rule for Disjoint Events: If A and B are disjoint events, then P(A or B) = P(A) + P(B).

Comment

When dealing with probabilities, the word “or” will always be associated with the operation of addition; hence the name of this rule, “The Addition Rule.”

Example

Recall the blood type example:

Blood TypeOABAB
Probability0.440.420.100.04

Here is some additional information:

* A person with type A can donate blood to a person with type A or AB.

* A person with type B can donate blood to a person with type B or AB.

* A person with type AB can donate blood to a person with type AB only.

* A person with type O blood can donate to anyone.

What is the probability that a randomly chosen person is a potential donor for a person with blood type A?

From the information given, we know that being a potential donor for a person with blood type A means having blood type A or O. We therefore need to find P(A or O). Since the events A and O are disjoint, we can use the addition rule for disjoint events to get: P(A or O) = P(A) + P(O) = 0.42 + 0.44 = 0.86. It is easy to see why adding the probability actually makes sense. If 42% of the population has blood type A and 44% of the population has blood type O, then 42% + 44% = 86% of the population has either blood type A or O, and thus are potential donors to a person with blood type A. This reasoning about why the addition rule makes sense can be visualized using the pie chart below:

A pie chart titled "Blood Types." Type A takes up 42% of the pie chart, and type O takes up 44%. Together, as A or O, they take up 86% of the pie chart.

P(A and B) for Independent Events

Rule 4, the addition rule, deals with finding P(A or B). We are now moving on to rule 5, which deals with yet another situation of frequent interest, finding P(A and B), the probability that both events A and B occur. In other words,

P(A and B) = P(event A occurs and event B occurs)

For example, we might be interested in the probability that if two people are chosen at random, both the first has blood type O and the second has blood type O. Since a person with blood type O can donate blood to anyone, this probability might be of particular interest in this context.

Using a Venn diagram, we can visualize “A and B,” which is represented by the overlap between events A and B:

A rectangle represents the entire sample space. Inside this rectangle are two circles, one for the occurrences in A and one for the occurrences in B. These two circles partially overlap. The area in the overlap contains the ocurrences for A and B.

Comment

There is one special case for which we know what P(A and B) equals without applying any rule.

So, if events A and B are disjoint, then (by definition) P(A and B)= 0. But what if the events are not disjoint?

Recall that rule 4, the Addition Rule, has two versions. One is restricted to disjoint events, which we’ve already covered, and we’ll deal with the more general version later in this module. The same is true of rule 5. Rule 5 has two versions. The version we’ll present here is restricted to a special case that we’ll now discuss, and there is a more general version that we’ll present in the next module.

The version of rule 5 that will be presented here applies to the special case in which the two events are independent of each other.independent(definition)

Two events A and B are said to be independent if the fact that one event has occurred does not affect the probability that the other event will occur. If whether or not one event occurs does affect the probability that the other event will occur, then the two events are said to be dependent.

Comment

It is quite common for students to initially get confused about the distinction between the idea of disjoint events and the idea of independent events. The purpose of this comment (and the activity that follows it) is to help students develop more understanding about these very different ideas.

The idea of disjoint events is about whether or not it is possible for the events to occur at the same time (see the examples on page 3 of the Probability Rules section).

The idea of independent events is about whether or not the events affect each other in the sense that the occurrence of one event affects the probability of the occurrence of the other (see the examples above).

The following activity deals with the distinction between these concepts.

The purpose of this activity is to help you strengthen your understanding about the concepts of disjoint events and independent events, and the distinction between them.

In each of the following questions, you are presented with a random experiment and two events related to it. You are asked to decide whether the events are disjoint or not, and whether the events are independent or not.

 Multiplication Rule for Independent Events

Now that we understand the idea of independent events, we can finally get to rule 5. As mentioned before, Rule 5 actually has two versions, one for finding P(A and B) in the special case in which the events A and B are independent, and a more general version for use when the events are not necessarily independent. We will first present the version of rule 5 that is restricted to independent events, and in the next section we will revisit Rule 5 and present the more general version.

Rule 5: The Multiplication Rule for Independent Events

If A and B are two independent events, then P(A and B) = P(A) * P(B).

Comment

When dealing with probabilities, the word “and” will always be associated with the operation of multiplication; hence the name of this rule, “The Multiplication Rule.”

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