What is a contingency table?

A contingency table organizes the joint probability distribution of two events in a grid. Interior cells show joint probabilities P(A ∩ B), row and column totals show marginal probabilities P(A) and P(B), and conditional probabilities can be calculated from these values.

How do you read conditional probability from the table?

Conditional probability P(A|B) equals the joint probability P(A ∩ B) divided by the marginal P(B). The table displays conditional probabilities in panels on the right. Click any row to see the numerator (joint cell) and denominator (marginal) highlighted.

What do the three sliders control?

P(A) sets the marginal probability of event A. P(B|A) sets the conditional probability of B given A. P(B|Aᶜ) sets the conditional probability of B given not-A. All six joint and marginal probabilities calculate automatically from these three inputs.

How does the table demonstrate Bayes' theorem?

The same joint probability P(A ∩ B) appears in both P(A|B) and P(B|A), but with different denominators (P(B) vs P(A)). Click P(A|B) then P(B|A) to see how Bayes' theorem relates these conditional probabilities through the shared joint probability.

What is the difference between joint and marginal probability?

Joint probability P(A ∩ B) is the probability both events occur together (interior cells). Marginal probability P(A) or P(B) is the probability of one event regardless of the other (row/column totals). Marginals equal the sum of joint probabilities in that row or column.

Contingency Table Visualization

How to read: The main table shows joint probabilities P(A∩B) in each cell. Row and column totals show marginal probabilities. The four tables on the right show all conditional probability distributions with their calculations. Click any conditional probability to highlight the corresponding cells in the main table.

Joint Probability Table

	B	B^c	Total
A	0.4200 P(A ∩ B)	0.1800 P(A ∩ B^c)	0.6000
A^c	0.1200 P(A^c ∩ B)	0.2800 P(A^c ∩ B^c)	0.4000
Total	0.5400	0.4600	1.0000

P(A) = 0.600

P(B|A) = 0.700

P(B|A^c) = 0.300

No selection made. Click any cell in the main table to see its joint probability explanation, or click any row in the conditional panels to understand how probabilities change when conditioning on known information.

Conditional on A

Click rows to highlight paths

P(B\|A) = P(A∩B) / P(A) = 0.42 / 0.60	0.7000
P(B^c\|A) = P(A∩B^c) / P(A) = 0.18 / 0.60	0.3000
Total	1.0000

Conditional on A^c

Click rows to highlight paths

P(B\|A^c) = P(A^c∩B) / P(A^c) = 0.12 / 0.40	0.3000
P(B^c\|A^c) = P(A^c∩B^c) / P(A^c) = 0.28 / 0.40	0.7000
Total	1.0000

Conditional on B

Click rows to highlight paths

P(A\|B) = P(A∩B) / P(B) = 0.42 / 0.54	0.7778
P(A^c\|B) = P(A^c∩B) / P(B) = 0.12 / 0.54	0.2222
Total	1.0000

Conditional on B^c

Click rows to highlight paths

P(A\|B^c) = P(A∩B^c) / P(B^c) = 0.18 / 0.46	0.3913
P(A^c\|B^c) = P(A^c∩B^c) / P(B^c) = 0.28 / 0.46	0.6087
Total	1.0000

Getting Started with the Contingency Table

Using the Probability Sliders

The Table at Its Opening Settings

Reading the Joint Probability Table

Understanding Marginal Probabilities

Exploring Conditional Probability Panels

Clicking to Trace Calculations

Bayes' Theorem in the Table

Related Tools and Concepts

Visualizing Conditional Probability with Contingency Tables

This interactive 2×2 contingency table displays the complete probability relationship between two events. Three sliders control P(A), P(B|A), and P(B|Aᶜ), with all joint, marginal, and conditional probabilities updating in real time. Click any cell or conditional probability row to highlight the mathematical relationship.

Getting Started with the Contingency Table

This interactive 2×2 contingency table displays the complete probability relationship between two events A and B. The table organizes joint probabilities, marginal probabilities, and conditional probabilities in a single unified view.

The left side shows the main probability table with four interior cells (joint probabilities) and row/column totals (marginal probabilities). The right side displays four conditional probability panels showing P(B|A), P(B|Aᶜ), P(A|B), and P(A|Bᶜ).

Three sliders below the table control P(A), P(B|A), and P(B|Aᶜ). All other probabilities derive from these three inputs. Click any cell or conditional probability row to highlight the mathematical relationship between values.

Using the Probability Sliders

Three sliders control the entire probability structure:

• P(A) sets the marginal probability of event A. This determines the row totals—how probability splits between row A and row Aᶜ.

• P(B|A) sets the conditional probability of B given A occurred. This determines how the A row splits between columns B and Bᶜ.

• P(B|Aᶜ) sets the conditional probability of B given A did not occur. This determines how the Aᶜ row splits between columns.

As you adjust any slider, all six interior cells and four marginal totals recalculate instantly. The conditional probability panels on the right also update. Try setting P(B|A) = P(B|Aᶜ) to see what happens—this creates independence between A and B.

The Table at Its Opening Settings

The sliders start at

P(A) = 0.6

P(B \mid A) = 0.7

and

P(B \mid A^c) = 0.3

, giving the four joint cells

0.42

0.18

0.12

and

0.28

.

Read across for the row marginals

0.60

and

0.40

; read down for the column marginals

0.54

and

0.46

. The grand total is exactly

1

The 2x2 table at P(A) = 0.6, P(B|A) = 0.7, P(B|not A) = 0.3

Cells 0.42 / 0.18 / 0.12 / 0.28, row totals 0.60 and 0.40, column totals 0.54 and 0.46, grand total exactly 1. P(B) = 0.54 is just a column sum.

The table and the tree are the same three numbers in two layouts, and each makes a different thing easy. The tree shows how the joints are *built* — multiply along a path. The table shows how the marginals are *recovered* — sum along a row or a column.

The column marginal is where the table earns its place.

P(B) = 0.42 + 0.12 = 0.54

is the law of total probability, and in this layout it is not a formula at all, just the total of a column.

Both conditionals live here too, as the same cell over two different totals:

P(B \mid A) = \frac{0.42}{0.60} = 0.7

divides by its row, and

P(A \mid B) = \frac{0.42}{0.54} = 0.7778

divides by its column. Bayes' theorem is the instruction to switch which total you divide by.

Reading the Joint Probability Table

The 2×2 table interior contains four joint probabilities:

• P(A ∩ B) — upper left: probability both A and B occur
• P(A ∩ Bᶜ) — upper right: probability A occurs but B does not
• P(Aᶜ ∩ B) — lower left: probability B occurs but A does not
• P(Aᶜ ∩ Bᶜ) — lower right: probability neither occurs

Each cell has a distinct color for visual tracking. Click any cell to highlight it and see how it appears in conditional probability calculations on the right.

The four joint probabilities sum to exactly 1.0000, representing the complete sample space. The table displays this sum in the bottom-right total cell.

Understanding Marginal Probabilities

Marginal probabilities appear in the "Total" row and column:

Row totals (right side):
• P(A) — sum of joint probabilities in the A row
• P(Aᶜ) — sum of joint probabilities in the Aᶜ row

Column totals (bottom):
• P(B) — sum of joint probabilities in the B column
• P(Bᶜ) — sum of joint probabilities in the Bᶜ column

The relationship:

P(A) = P(A \cap B) + P(A \cap B^c)

P(B) = P(A \cap B) + P(A^c \cap B)

Row marginals sum to 1: P(A) + P(Aᶜ) = 1. Column marginals also sum to 1: P(B) + P(Bᶜ) = 1. Marginal probabilities serve as denominators in conditional probability formulas.

Exploring Conditional Probability Panels

Four panels on the right display conditional probability distributions:

Top row panels condition on A:
• P(B|A) and P(Bᶜ|A) — given A occurred, probability of B vs Bᶜ
• P(B|Aᶜ) and P(Bᶜ|Aᶜ) — given A did not occur, probability of B vs Bᶜ

Bottom row panels condition on B:
• P(A|B) and P(Aᶜ|B) — given B occurred, probability of A vs Aᶜ
• P(A|Bᶜ) and P(Aᶜ|Bᶜ) — given B did not occur, probability of A vs Aᶜ

Each conditional probability shows its formula breakdown. Click any row to highlight the numerator (joint cell) and denominator (marginal total) in the main table, visually confirming:

P(A|B) = \frac{P(A \cap B)}{P(B)}

Clicking to Trace Calculations

The highlighting system reveals how conditional probabilities derive from the table:

Click a joint cell (e.g., P(A ∩ B)):
The cell highlights in its unique color. Look at the conditional panels to see where this joint probability appears as a numerator.

Click a conditional probability row (e.g., P(A|B)):
Three things highlight simultaneously:
• The joint probability cell (numerator)
• The marginal probability (denominator)
• The formula showing the division

This visualization demonstrates that conditional probability is always a ratio: the intersection divided by the condition. The same joint probability P(A ∩ B) appears in both P(A|B) (denominator P(B)) and P(B|A) (denominator P(A)).

Use the Clear Selection button to remove all highlighting.

Bayes' Theorem in the Table

The contingency table demonstrates Bayes' theorem visually. Compare P(A|B) with P(B|A):

Both use the same numerator P(A ∩ B), but different denominators:

P(A|B) = \frac{P(A \cap B)}{P(B)} \quad \text{vs} \quad P(B|A) = \frac{P(A \cap B)}{P(A)}

Bayes' theorem relates these:

P(A|B) = \frac{P(B|A) \times P(A)}{P(B)}

Try this: click P(A|B), note which cells highlight. Then click P(B|A). Both highlight the same green joint cell but with different marginal denominators. This is Bayes' theorem in action—flipping the condition changes the denominator.

The table shows P(B) calculated via total probability: P(B) = P(A)·P(B|A) + P(Aᶜ)·P(B|Aᶜ), which appears in Bayes' formula's denominator.

Contingency Table Visualization

Joint Probability Table

Conditional on A

Conditional on Ac

Conditional on B

Conditional on Bc

Visualizing Conditional Probability with Contingency Tables

Getting Started with the Contingency Table

Using the Probability Sliders

The Table at Its Opening Settings

Reading the Joint Probability Table

Understanding Marginal Probabilities

Exploring Conditional Probability Panels

Clicking to Trace Calculations

Bayes' Theorem in the Table

Related Tools and Concepts

Conditional on A^c

Conditional on B^c