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Section 5.3: Independence and the Multiplication Rule


By the end of this lesson, you will be able to...

  1. identify independent events
  2. use the Multiplication Rule for independent events
  3. compute "at least" probabilities

For an overview probability, watch this video:


One of the most important concepts in probability is that of independent events.

Two events E and F are independent if the occurrence of event E does not affect the probability of event F.

Let's look at a couple examples.

Example 1
playing cards
Source: stock.xchng

Consider the experiment where two cards are drawn without replacement. (Without replacement means one is drawn and then the second is drawn without putting the first one back.) Define events E and F this way:

E = the first card drawn is a King

F = the second card drawn is a King

Are events E and F independent?

[reveal answer]

Definitely not. P(F) depends on what happens with E. If a King is drawn with the first card, P(F) = 3/51, but if a King is not drawn, P(F) = 4/52 = 1/13.

Example 2

Consider the experiment in which two fair six-sided dice are rolled, and define events E and F as follows:

E = the first die is a 3

F = the second die is a 3

Are events E and F independent?

[reveal answer]

Yes. In this case, what happens on the first die does not affect what happens on the second die. P(F) = 1/6, regardless of what happens with the first die.

Disjoint vs. Independent

It is very common for students to confuse the concepts of disjoint (mutually exclusive) events with independent events. Recall from the last section:

Two events are disjoint if they have no outcomes in common. (Also commonly known as mutually exclusive events.)

Here's a Venn diagram of two disjoint events.

disjoint events

Looking at this image, we can see very clearly that if event E occurs (that is, the outcome outcome is in event E), it cannot possibly be in event F. So E and F are dependent, since the occurrence of event E made event F impossible (i.e. the occurrence of event E changed the probability of event F).

The Multiplication Rule for Independent Events

To introduce the next idea, let's look at the experiment from Example 2, in Section 5.1.

Example 3

The experiment was rolling a fair six-sided die twice. Suppose define the event E:

E = both dice are 2's

two dice

The possible outcomes are (1,1), (1,2), (1,3), ... (6,5), and (6,6). Since only one of these is (2,2), we know P(E) = 1/36. Let's look at it another way, though.

P(E) =  N(E)  =  1  =  1  =  1 1  = P(E)•P(F) 
N(S) 36 6•6 6 6

In fact, this will always be true if E and F are independent.

Multiplication Rule for Independent Events

If E and F are independent events, then

P(E and F) = P(E)•P(F)

Example 4

According to data from the American Cancer Society, about 1 in 3 women living in the U.S. will have some form of cancer during their lives.

If three women are randomly selected, what is the probability that they will all contract cancer at some point during their lives?

[reveal answer]

Since the three women are randomly selected, we can assume that they are independent of each other. Because of that, we can use the Multiplication Rule for Independent Events:

P(all have breast cancer)
= P(1st does and 2nd does and 3rd does)
= P(1st) P(2nd) P(3rd)
= (1/3)(1/3)(1/3)
≈ 0.037

So there is about a 3.7% probability that all 3 of the women will contract cancer at some point.

At-least Probabilities

The phrase "at least" can make a seemingly simple problem much more difficult. For example, suppose we're looking at cancer rates in women. And suppose we have a random sample of 5 women. If we're looking for the probability that at least one will have some form of cancer, that's really:

P(1 will have cancer) + P(2 will have cancer) + ... + P(all 5 will have cancer)

Instead, it's much easier to use the Complement Rule, from Section 5.2.

The Complement Rule

P(E) + P(Ec) = 1

the complement of E

In our example, the complement of at least one will have cancer is none will have cancer. So P(at least one will have cancer) = 1 - P(none will have it) = 1 - (2/3)5 ≈ 0.8683.

Much easier! Keep this idea of at least probabilities and the Complement Rule in mind when you're looking at cases like this.