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Probability Diagrams

Probability

Reading a probability off

  • A Venn diagram may show the individual items, or just how many fall in each region
  • Either way, the probability is the count in the regions you want divided by the total count
  • Fill in the overlap first, then work outwards, because the numbers given usually include the overlap
    • If 18 play football, 14 play tennis and 7 play both, then 11 play only football and 7 play only tennis
A Venn diagram of two sets S and G inside a universal set: 12 in S alone, 3 in the overlap, 8 in G alone, and 7 outside both circles, so the total is 30
Source: Probability & Venn diagrams by Save My Exams
  • The region outside both circles holds everything in neither, and it is easy to forget when totalling
  • The notation P(A ∩ B) for "both" and P(A ∪ B) for "either" may be used
The same Venn diagram with only the part of S outside G shaded, so the 12 in that region is the numerator for the probability of being in S but not in G, over a total of 30
Source: Probability & Venn diagrams by Save My Exams
Exam tip

Every one of the 7 papers with a tree diagram asks you to complete it first, so fill the branches in before calculating anything, and leave fractions unsimplified while you work so they add easily. Without replacement, the denominator drops by one and the numerator drops only for what was actually taken. For "at least one", subtract the opposite case from 1. These papers say "sample space diagram" — "possibility diagram" and "two-way table" appear nowhere.

Worked example

Probabilities from a Venn diagram

In a group of 30 students, 18 play football, 14 play tennis and 7 play both. A student is chosen at random. Find the probability that the student plays neither sport, and the probability that a student who plays football also plays tennis.

Solution:

  • Start with the overlap: 7 play both
  • Football only = 18 − 7 = 11, and tennis only = 14 − 7 = 7
  • Those three regions hold 11 + 7 + 7 = 25 students
  • So the number playing neither is 30 − 25 = 5
  • P(neither) = 5/30 = 1/6
  • For the second part, only football players are being considered, so the total is 18, not 30
  • Of those 18, the ones who also play tennis are the 7 in the overlap
  • P(tennis, given football) = 7/18