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Introduction to Probability

Previously, we used a two-way table to examine the association between two categorical variables. Now we will do the same calculations through a new lens; the lens is probability. Later in the course we will study probability in more detail because it is at the heart of statistical inference, but this is a natural place to begin to use probability language.

We use the word probability to mean "likelihood" or "chance." For our purposes, probability describes the chance of a specific outcome if we are randomly selecting an individual from a group. When we ask questions such as "who is more likely to feel that they are overweight, a male or a female college student?", we are using probability language. Similarly, the question, "what percentage of female college students feel that they are overweight" is the same as asking, "what is the probability that female college student feels she is overweight?" We do the same calculations to answer each of these questions.

EXAMPLE Using a two-way table to answer probability questions

Let's revisit the data from the online health behavior survey sent to 25,000 college students in 2009 and use it to calculate probabilities. Of those surveyed, 6,055 students responded.

What is the probability that a Health Science major smoked in the last month?

The calculation is the same as a conditional percentage:
Of the 944 Health Science majors, 227 smoked in the last month. 227/944 = 0.24.

The interpretation:
Based on this data, the probability that a Health Science major smoked in the last month is 24%. We can also say, there is a 24% chance that a randomly selected Health Science major smoked in the last month.

Who is more likely to have smoked in the last month, a Health Science major or a Humanities major?

We compare two conditional percentages to answer this question. We will interpret each conditional percentage as a probability.

We already calculated the likelihood that a Health Science major smoked in the last month, so let's do the same for the Humanities major.

The calculation: Of the 651 Humanities majors, 197 smoked in the last month. (197/651) = 0.30 = 30%

The interpretation: Compared to Health Science majors, Humanities majors are more likely to have smoked in the last month. There is a 30% chance that a randomly selected Humanities major smoked in the last month compared to a 24% chance for a Health Science major.

The Relationship of Field of Study to Current Smoking Status among College Students

| Smoked in Last 30 Days | Did Not Smoke in Last 30 Days | Row Totals
Art, design, performing arts | 149 | 336 | 485
Humanities | 197 | 454 | 651
Communication, languages | 233 | 389 | 622
Education | 56 | 170 | 226
Health Sciences | 227 | 717 | 944
Math, engineering, sciences | 245 | 924 | 1,169
Social science, human services | 306 | 593 | 899
Independent study | 134 | 260 | 394
Undeclared | 176 | 489 | 665
Column Totals | 1,723 | 4,332 | 6,055

(C. J. Berg, C. M. Klatt, J. L. Thomas, J. S. Ahluwalia, and L. C. An, "The Relationship of Field of Study to Current Smoking Status among College Students," College Student Journal 43(3):744–754, 2009.)

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