Conditional probability

Conditional probability is the probability of A when we already know that B happened. It is written P(A | B), read "A given B".

P(A | B) = P(A and B) ÷ P(B)

Knowing B shrinks the world to the outcomes in B; we then ask how many of those are also in A.

From a table of counts

100 students: 40 study daily, and 30 of those pass; of the 60 who do not, 24 pass.

P(pass | study daily) = 30 ÷ 40 = 0.75, but P(pass | do not) = 24 ÷ 60 = 0.4. Studying and passing are not independent.

Without replacement

Draw two cards. P(first is an ace) = 4/52. If it was, 3 aces remain among 51 cards: P(second is an ace | first was) = 3/51. Both are aces with probability 4/52 × 3/51.

from fractions import Fraction

pass_if_study = Fraction(30, 40)
pass_if_not = Fraction(24, 60)
print("P(pass | study) =", pass_if_study)
print("P(pass | not) =", pass_if_not)

first = Fraction(4, 52)
second_given_first = Fraction(3, 51)
print("P(two aces) =", first * second_given_first)

Common mistakes

• Confusing P(A | B) with P(B | A). "Most people who pass study daily" is not the same as "most people who study daily pass". The next lesson is about exactly this.

Mark lesson complete