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Conditional Probability and Contingency Tables

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This lesson will demonstrate finding conditional probabilities from a contingency table.

Tutorial

This tutorial will cover conditional probability in the context of two-way tables. You will learn about:

- Conditional Probability
- Tables

You can use two-way tables to find conditional probabilities.

Conditional probability is the probability that some event (B) occurs given that some other event (A) has already occurred. It's written this way, probability of B given A:

Suppose that middle school students were asked which was their dominant hand. Here the results are shown in the two-way table:

If a student is a sixth grade, what's the probability that he or she is left-handed? To find the answer, isolate the sixth grade row.

This is a question of conditional probability: the probability of a student being left-handed given that the student is a sixth grader. The formula looks like this:

This formula shows that probability of L and 6, left hand and sixth grade, over the probability of sixth grade. So the probability that a student is left-handed and a sixth grader is 9 out of the 338 middle schoolers. The probability that a student is in sixth grade is 110 out of the 338. That reduces to 9 out of 110.

Notice that you can use the probabilities, which were both divided by 338, the grand total. Or you can just use the frequencies from the cells and from the marginal distributions in the column or row totals.

What is the probability that a left-handed student is in sixth grade? At first glance, this looks like the same question: what's the probability that a student is in sixth grade given that they are a lefty? So it's the same probability of L and 6.

But in this question, the denominator is different, or probability of L. Probability of L and 6, the frequency there was 9. The lefties, there were 51 of them. So the answer is 9 out of 51.

What is the probability that a seventh grade student is ambidextrous? And what is the probability that a student is right-handed given that he or she is an eighth grader?

You likely came up with this:

You can use a two-way table that actually has probabilities in it, or relative frequencies. Here is an example of how that would look:

This shows that 5% of all of these kids are boys that enjoy cheese pizza. 12% of all of the kids are boys that enjoy pepperoni, et cetera.

To find the probability of a student preferring cheese pizza given that he's a boy, you can use the same with the marginal distributions in the row totals and the column totals, but use the probabilities instead of the frequencies.

So the probability that a student enjoys cheese and is a boy is the 0.05 value from the table. And the probability of being a boy, in this particular sample, 36% of the sample were boys. And so that reveals that there's about 14% probability that if you are a boy that you'll prefer cheese pizza.

**Conditional probability** is the probability that some other event follows some other event which has already occurred. It's calculated by dividing the joint probability, probability of A and B, by the probability of the event which has already occurred. In most cases, we were using the probability of A. This formula works for all events, not just for independent events or mutually exclusive events, and the data for these formulas can be found in **two-way tables**.

Thank you and good luck!

Source: THIS WORK IS ADAPTED FROM SOPHIA AUTHOR JONATHAN OSTERS