In this section we learn about events that are not independent of one another. When this happens, knowing additional information can actually change the probability of a future event happening. How can this occur? Aren’t probabilities supposed to be fixed?

The easiest example deals with dice. Let’s suppose you close your eyes and roll a die. Without opening your eyes, what is the probability that you rolled the number 5? That’s easy,

Let’s change up the experiment a bit. You close your eyes and, after rolling your die, a friend in the room tells you that you rolled an odd number. Now, what is the probability that you rolled the number 5? Since there are only three odd numbers on a die, {1,3,5}, you now have a 1 in 3 chance of rolling a 5. In symbols:

If your friend tells you that an even number showed up, what is the probability that you rolled a 5? It can’t happen since 5 is an odd number.

So what is happening in these cases? Well, you are learning some additional information that leads us to change the probability of an event occurring. In effect, knowing additional information changes the sample size we use to compute the probabilities. Therefore, the probability of our event occurring must change.

The notation *P*(*F*│*E*) means “the probability of *F* occurring given that (or knowing that) event *E* already occurred.” For the above dice example, *F* = {roll a 5}, and *E *= {result is an odd number}, and we found that *P*(*F*│*E*) = 33.33%.

Conditional probabilities are useful when presented with data that comes in tables, where different **categories** of data (say, Male and Female), are broken down into additional sub-categories (say, marriage status).

To compute the probabilities of dependent data, we use the *Conditional Probability Rule*. In symbols:

where *P*(*E*) is the probability of event *E* occurring and *N *(*E*) is the number of ways that event *E* can occur.

##### Example

Consider studying the possibilities of gender for a 2-child family. The sample space for all possible outcomes is *S *= {*GG, GB, BG, BB*}, where birth order is important…there is a first child and then a second child. Assume that each child is equally likely to be male or female. Each of the items in our sample space can be thought of as the outcome of a chance experiment that selects at random a family with two children. Think about the following questions:

- What is the probability of seeing a family with two girls, given that the family has at least one girl?
- What is the probability of seeing a family with two girls, given that the older sibling is a girl?

To most people, these questions seem to be the same. However, if we fill in the probabilities you’ll see they are different!

**For Question 1**, we want to compute:

*P *(family has two girls | family has an older girl)

Using the Conditional Probability Rule we see this probability is equal to:

**For Question 2**, we want to compute:

*P* (family has two girls | family has an older girl)

Using the Conditional Probability Rule we see this probability is equal to:

Did you notice how the sample space for Question 2 changed? Since we knew the older child was a girl, we had to eliminate the outcome of {*BG*} since this family had a boy first.

#### Computing Probabilities Using the General Multiplication Rule

Earlier you saw the multiplication rule for *independent events*, which is:

*P* (*E*** and** *F*) = *P* (*E*) ⋅ *P* (*F*)

Is there such a rule if events *E* and *F* are *dependent*? With a slight modification, we get the *General Multiplication Rule*:

*P *(*E* **and** *F*) = *P *(E) ⋅ *P *(*F*│*E*)