0580

Conditional Probability

Probability

Narrowing the total

  • Identify the condition first and shade or ring the region it describes
  • Count how many are in that region: this becomes the denominator
  • Then count how many of those also satisfy the thing being asked for: this becomes the numerator
  • The overall total is not used at all, which is exactly what makes it conditional
  • The same method works on a two-way table, where the condition picks out one row or one column and the total for that row or column is the denominator
A Venn diagram listing individual numbers rather than counts: 6, 12 and 2 lie in A alone, 14 and 28 in the overlap, 7, 21 and 35 in B alone, and 8, 5 and 1 outside both, so a condition such as being in A restricts the total to the five numbers inside A
Source: Conditional Probability by Save My Exams
Worked example

A conditional probability from a Venn diagram

Of 40 people, 22 own a car, 16 own a bicycle and 9 own both. One person is chosen at random. Find the probability that a person who owns a car also owns a bicycle, and the probability that a person who owns a bicycle also owns a car.

Solution:

  • Fill the overlap first: 9 own both
  • Car only = 22 − 9 = 13, and bicycle only = 16 − 9 = 7
  • Those regions hold 13 + 9 + 7 = 29, so 40 − 29 = 11 own neither
  • For the first part the condition is owning a car, so the total is the 22 car owners, not 40
  • Of those 22, the ones who also own a bicycle are the 9 in the overlap
  • P(bicycle, given car) = 9/22
  • For the second part the condition is owning a bicycle, so the total is now 16
  • P(car, given bicycle) = 9/16
  • The two answers differ, which shows the order of the condition matters