Two-Way Tables (3 of 5)

Learning Objectives

• Calculate marginal, joint, and conditional percentages and interpret them as probability estimates.

At this point, we know how to determine marginal probabilities, such as the probability that a randomly selected student is female: P(female).

And we know how to calculate conditional probabilities, such as the probability that a randomly selected female student is in the Health Science program: P(Health Science | female)

But we do not know how to calculate joint probabilities, such as the probability that a randomly selected student is both a female and in the Health Sciences program.

We write this joint probability as P(female and Health Sciences).

The following example illustrates how to calculate a joint probability.

Joint Probability

Question: If we select a student at random, what is the probability that the student is both a male and in the Info Tech program?

Answer: This question involves male students who are in the Info Tech program, but it is NOT a conditional probability. We are picking a student at random from the entire population of 12,000 students, so there is no condition. Our shorthand notation for this probability is:

• P(male and Info Tech)

Since 564 of the 12,000 students enrolled at the college are both male and in the Info Tech program (see table), the probability P(male and Info Tech) is:

$\frac{\text{564}}{\text{12,000}}\approx \text{.05}$

 Arts-Sci Bus-Econ Info Tech Health Science Graphics Design Culinary Arts Row Totals Female 4,660 435 494 421 105 83 6,198 Male 4,334 490 564 223 97 94 5,802 Column Totals 8,994 925 1,058 644 202 177 12,000

We call this calculation a joint probability. Note that when we calculate a joint probability, we divide the count from an inner cell of the table by the overall total count in the lower right corner.

The following table is used for the next Learn By Doing and Did I Get This? activities.

 Arts-Sci Bus-Econ Info Tech Health Science Graphics Design Culinary Arts Row Totals Female 4,660 435 494 421 105 83 6,198 Male 4,334 490 564 223 97 94 5,802 Column Totals 8,994 925 1,058 644 202 177 12,000