Definition: Binary contingency table
Definition: Let $A$ and $B$ be random events and let $\overline{A}$ and $\overline{B}$ denote their complements. Then, (i) the set of relative frequencies of $A$’s and $B$’s occurence or (ii) the set of absolute frequencies of the combinations of $A$/$\overline{A}$ and $B$/$\overline{B}$ in a sample is referred to as a $2 \times 2$ contingency table.
Observed data from a $2 \times 2$ contingency table are characterized as follows:
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$y_1 = y_{11}$ is the number of cases in which $A$ and $B$;
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$y_2 = y_{10}$ is the number of cases in which $A$ and $\overline{B}$;
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$y_3 = y_{01}$ is the number of cases in which $\overline{A}$ and $B$;
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$y_4 = y_{00}$ is the number of cases in which $\overline{A}$ and $\overline{B}$;
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$y_{1 \bullet} = y_{11} + y_{10}$ is the (marginal total) number of cases in which $A$;
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$y_{0 \bullet} = y_{01} + y_{00}$ is the (marginal total) number of cases in which $\overline{A}$;
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$y_{\bullet 1} = y_{11} + y_{01}$ is the (marginal total) number of cases in which $B$;
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$y_{\bullet 0} = y_{10} + y_{00}$ is the (marginal total) number of cases in which $\overline{B}$;
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$n = y_{11} + y_{10} + y_{01} + y_{00}$ is the total number of cases.
The distribution underlying a $2 \times 2$ contingency table is characterized as follows:
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$p_1 = p_{11}$ is the joint probability that $A$ and $B$;
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$p_2 = p_{10}$ is the probability that $A$ and $\overline{B}$;
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$p_3 = p_{01}$ is the probability that $\overline{A}$ and $B$;
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$p_4 = p_{00}$ is the probability that $\overline{A}$ and $\overline{B}$;
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$p_{1 \bullet} = p_{11} + p_{10}$ is the marginal probability that $A$;
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$p_{0 \bullet} = p_{01} + p_{00}$ is the marginal probability that $\overline{A}$;
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$p_{\bullet 1} = p_{11} + p_{01}$ is the marginal probability that $B$;
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$p_{\bullet 0} = p_{10} + p_{00}$ is the marginal probability that $\overline{B}$;
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$1 = p_{11} + p_{10} + p_{01} + p_{00}$ is the probability of the sample space.
- Wikipedia (2026): "Contingency table"; in: Wikipedia, the free encyclopedia, retrieved on 2026-09-11; URL: https://en.wikipedia.org/wiki/Contingency_table#Example.
Metadata: ID: D239 | shortcut: ct2x2 | author: JoramSoch | date: 2026-09-11, 09:58.