Statistics

Venn diagrams

Explore overlapping events and conditional probability.

Predict, test and explain

Step 1 of 3

Predict

There are 3 members in both A and B and 2 in B only. Given that the member is in B, what is P(A)?

P(A\cap B)=\frac{1}{4}

Scroll the diagram sideways to see every label.

P(A)=\frac{7}{12}

P(B)=\frac{5}{12}

P(A\cup B)=\frac{3}{4}

P(A\mid B)=\frac{3}{5}

Choose one of 12 equally likely members. Here P(A and B) differs from P(A) × P(B).

Venn regions: 12 equally likely members
Region	Members	Probability
A only	4	$\frac{1}{3}$
Both A and B	3	$\frac{1}{4}$
B only	2	$\frac{1}{6}$
Neither event	3	$\frac{1}{4}$

Explore and compare

Choose the investigation above. Predict what will change before you move a control.

Predict

Try it in the simulation. Change one thing at a time.

Explain

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Hints, self-check and connections

Try a self-check

Fixed values, separate from the diagram controls.

A Venn diagram has 3 in A only, 4 in both, 6 in B only and 2 in neither. One B member is chosen uniformly. Find the probability they also belong to A.

$P(A\mid B)=?$

Hint

Restrict the total to the 10 members of B.

Reveal answer

Four of the ten B members are also in A, so the probability is two fifths.

$P(A\mid B)=\frac{4}{4+6}=\frac25$

Build an explanation

Hint 1

Count the overlap once when finding the total number of members.

Hint 2

A includes A-only and the overlap; B includes B-only and the overlap.

Hint 3

For probability given B, use the B members as the new total and count those also in A.

Worked example

This example uses fixed values, separate from the diagram controls.

There are 4 in A only, 3 in both, 2 in B only and 3 in neither. Find P(A given B).

B contains 3 + 2 = 5 members.
Of those five, three are also in A.
Divide the overlap by the B total.

Answer: 3/5.

Connect this idea

Watch out: Overlapping events are not automatically independent; conditional probability uses the restricted total.