Conditional Probability Venn Diagrams

# Conditional Probability Venn Diagrams

GCSE(H),

Venn diagrams are used to determine conditional probabilities. The conditional probability is given by the sets and intersections of these sets.

Conditional probability is based upon an event A given an event B has already happened: this is written as P(A | B).

The probability of A, given B, is the probability of A and B divided by the probability of A:

P(A) = frac(text(P)(A nn B))(text(P)(B))

In Venn diagrams, this is the intersection set divided by the set being considered.

## Examples

1. The Venn diagram shows students that are studying a Science subject. The Venn diagram shows those studying Biology and Chemistry. What is the probability of a student studying Biology if they are also studying Chemistry?

Answer: P(B | C) = frac(text(P)(B nn C))(text(P)(C)) = frac(13)(27)

Probability of Biology given Chemistry: P(B | C)

The number of students studying Biology and Chemistry = P(B nn C) = 13 students

The total number of students studying Chemistry = P(C) = 27

P(B | C) = frac(text(P)(B nn C))(text(P)(C)) = frac(13)(27)

2. The Venn diagram shows students that are studying a Language subject. What is the probability of a student studying three subjects, given that they are studying at least two subjects?

Answer: P(three languages | two languages) = frac(7)(22)

Taking at least two languages = 6 + 5 + 7 + 4 = 22

P(3lang | 2lang) = Pfrac(text(students taking 3 languages))(text(all students taking 2 or more languages)) = frac(7)(22)