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.

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)`

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

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)`

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

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