Introducing Conditional Probability

Mary Lynn is a senior at her school. After her three years of observing, she thinks that girls with blond hair are more likely to be shorter than girls with brown hair. She decides to do a project, and sample 40 girls to find out. She uses a random sample to get 25 girls with blond hair and 25 with brown hair. She defines “tall” as being 5’9”, and obtains each girl’s height. Her findings are in the contingency table below.

	Shorter than 5’9”	5’9” or Taller	Total
Blond	22	3	25
Brown	14	11	25
Total	36	14	50

A way to represent the 4 different possibilities of hair color and height is to use a tree diagram. This shows us the different ways the outcomes could occur, such as choosing a girl with blond hair who is shorter than 5’9”.

This tree diagram helps us see what the possible outcomes are when observing a particular girl in the sample. First, her hair is either blond or brown, and then we consider her height.

Conditional probability has to do with the question, “Given that a certain thing happens, what is the probability that something else happens in addition to that?”

The following questions explore conditional probabilities in this situation.

What percent of girls in the study were 5’9” or taller?

14/50 = .28 x 100 = 28 %

Shorter than 5’9”?

36/50 = .72 x 100 = 72 %

Of the girls with blond hair, what percent were shorter than 5’9”?

	Shorter than 5’9”	5’9” or Taller	Total
Blond	22	3	25
Brown	14	11	25
Total	36	14	50

First we look at the whole row of girls with blond hair; this is the total number of girls we are looking at right now (25). Next we look in the column for girls shorter than 5’9”. They intersect in the cell with value 22. So 22 out of 25 girls with blond hair were shorter than 5’9”. What percentage is this?

22/25 = .88 x 100 = 88 %

What percent were 5’9” or taller?

We do the same thing now, but we look at the 5’9” or taller column, which intersects the “blond” row at 3. So 3 out of 25 girls with blond hair were 5’9” or taller. Find the percentage:

3/25 = .12 x 100 = 12 %

So obviously, the majority of the girls who had blond hair were shorter than 5’9”.

Out of the girls who were 5’9” or taller, what percent of them had brown hair?

	Shorter than 5’9”	5’9” or Taller	Total
Blond	22	3	25
Brown	14	11	25
Total	36	14	50

So we look at the 5’9” or taller column, and that is the total number we are working with (14). Then we look at the brown hair row, and these two intersect at the cell with value 11.

So 11 out of 14 girls taller than 5’9” had brown hair. What percentage is this?

11/14 = .786 x 100 = 78.6 %

What was Mary Lynn ultimately trying to determine? She wanted to know if there was any association between hair color and height, i.e. she wanted to know if hair color and height are independent or dependent. Two events are independent if there is no association between them, i.e. if their conditional probabilities are equal to the marginal probabilities. A marginal probability is the probability obtained by taking the total number of one variable divided by the total number of items in the survey. Here’s what that would look like in our example:

We had an even split for hair color: 50% of the girls had blond hair, and 50% had brown hair. If height were independent from hair color, then we should have approximately 50% of the girls for each hair color shorter than 5’9”, and about 50% taller than 5’9”. So the proportions we get by taking the totals in the margins should be equal (marginal proportions). If the marginal proportions were equal, we would see something more like this:

	Shorter than 5’9”	5’9” or Taller	Total
Blond	12	13	25
Brown	14	11	25
Total	26	24	50

As we see in the above contingency table and calculations, if hair color and height were not associated, the proportion of girls who have blond hair should be equal to the even proportion in the right margin.

So are hair color and height independent? Let’s look back at the proportions we calculated earlier:

What percent of girls in the study were 5’9” or taller?

14/50 = .28 x 100 = 28 %

Shorter than 5’9”?

36/50 = .72 x 100 = 72 %

Since .28 and .72 are not at all close to .50, we see that there is a difference between the marginal proportions. This information could definitely lead us to suspect that there is some kind of association between hair color and height.

Additional Activity: “To Replace or Not to Replace?”

This activity allows students to use their graphing calculators to explore the scenarios of drawing marbles out of a bag, first replacing each marble after a draw, and then not replacing each marble. How does this affect the probability? This exploration will help students to understand even more about conditional probability and independent/dependent events.

More Applications of Conditional Probability

· Sampling with/without replacement: This is the concept dealt with in the TI activity. When drawing marbles from a bag, cards from a deck, or people out of a group, you can either replace each selection before drawing again, or not replace it. The probabilities will turn out differently if you replace vs. not replacing. If there are 3 red marbles in a bag of 5 marbles, the probability of drawing a red will be 3/5 each time if you replace, but it will decrease each time you draw a red marble if you do not replace it.

· Diagnostic testing: A common application of conditional probability is testing for drugs or diseases. When a test like this is given, the person being tested receives either a positive or negative test result, and either does or does not have the disease (or use the drugs). Many questions can be asked, such as:
- What is the probability you have the disease if you get a

positive test result? (Given that you receive a positive test result, what is the probability you have the disease? i.e. what is P(disease|positive)?)

- What is the probability you will get a negative test result if you have the disease? (Given that you have the disease, what is the probability you will receive a negative test result? i.e. what is P(negative|disease)?)

- What is the probability a person will test positive for using drugs if he/she does not actually use them?
(Given that a person does not use drugs, what is the probability they will receive a positive test result? i.e. what is P(positive|not drug user)?)

These last two questions deal with the concepts of false positive and false negative. A false negative occurs when a test result comes back negative, but it is false, i.e. a person actually does have that which he or she is being tested for. A false positive is the opposite: a person receives a positive test result when he or she actually does not have what he or she is being tested for. This concept relies heavily on conditional probability, and will be explored more in the next day’s lesson.