Type of probabilities in a contingency table

There are three types of probabilities for contingency table/crosstabs.

They are: Marginal probabilities, Joint probabilities and Conditional probabilities.

Suppose data is collected in a study of 165 students about their gender and insomnia status. After the collection of data, following 2×2 contingency table is generated:

Table 1: Contingency table of Insomnia by Gender

Marginal probabilities denote probabilities of a single variable. From the table 1, four marginal probabilities can be calculated. P (Male) = 103/165 = 0.62 It is the probability that a person selected at a random from the study is male. P (Female) = 62/165 = 0.38 It is the probability that a person selected at a random from the study is female. P (Insomniac) = 61/165 = 0.37 It is the probability that a person selected at a random from the study is insomniac. P (Non-insomniac) = 104/165 = 0.63 It is the probability that a person selected at a random from the study is non-insomniac.

Joint probabilities denote probabilities of intersection of both variables. From the table 1, four joint probabilities can be calculated. P (Male∩Insomniac) = 28/165 = 0.17 It is the probability that a person selected at a random from the study is male and insomniac. P (Male∩Non-insomniac) = 75/165 = 0.45 It is the probability that a person selected at a random from the study is male and non-insomniac. P (Female∩Insomniac) = 33/165 = 0.20 It is the probability that a person selected at a random from the study is female and insomniac. P (Female∩Non-insomniac) = 29/165 = 0.18 It is the probability that a person selected at a random from the study is female and non-insomniac.

Conditional probabilities denote probabilities of one variable at each level of another variable. From the table 1, eight conditional probabilities can be calculated, four of them are also called row proportions and four of them are also called column proportions. Row proportions: P (Insomniac | Male) = 28/103 = 0.27 It is the probability that a person selected at a random from the study is insomniac given that a person is male. P (Non-insomniac | Male) = 75/103 = 0.73 It is the probability that a person selected at a random from the study is non-insomniac given that a person is male. P (Insomniac | Female) = 33/62 = 0.53 It is the probability that a person selected at a random from the study is insomniac given that a person is female. P (Non-insomniac | Female) = 29/62 = 0.47 It is the probability that a person selected at a random from the study is non-insomniac given that a person is female. Column proportions: P (Male | Insomniac) = 28/61 = 0.46 It is the probability that a person selected at a random from the study is male given that a person is insomniac. P (Female | Insomniac) = 33/61 = 0.54 It is the probability that a person selected at a random from the study is female given that a person is insomniac. P (Male | Non-insomniac) = 75/104 = 0.72 It is the probability that a person selected at a random from the study is male given that a person is non-insomniac. P (Female | Non-insomniac) = 29/104 = 0.28 It is the probability that a person selected at a random from the study is female given that a person is non-insomniac.

Reference: Agresti A. An introduction to categorical data analysis. 2nd ed.: A John Wiley & Sons, Inc., Publication; 2007. p. 22