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Conditional Probability Venn Diagrams

Conditional Probability Venn Diagrams

Venn diagrams are used to determine conditional probabilities. The conditional probability is given by the 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) (probability of A given 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.

Conditional probability Venn Diagram

Example 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?

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)`

Conditional probability Venn diagram two languages

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

Example 2

The Venn diagram shows students that are studying a language subject (French, Spanish and German). What is the probability of a student studying three subjects, given that they are studying at least two subjects?

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

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

Conditional probabilty Venn diagram three languages

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