MathematicsExam code: 0580

Probability Diagrams

Probability

PractiseFlashcards

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
  • 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

Worked example

Probabilities from a Venn diagram

In a group of 30 students, 18 play football, 14 play tennis and 7 play both. Find the probability that a student chosen at random plays neither sport. Then two of the football players are chosen at random: what is the probability that exactly one of them 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
A rectangle labelled ℰ with two overlapping circles F and T. F only holds 11, the overlap 7, T only 7, and 5 lie outside both circles. The whole of circle F is shaded.
  • P(neither) = 5/30 = 1/6
  • For the second part, both students come from the football circle, so the first pick is out of 18, not 30
  • Of those 18, the 7 in the overlap play tennis and the 11 in football only do not
  • Two different students are chosen, so the second pick is out of the 17 who are left
  • Exactly one plays tennis means tennis then not, or not then tennis
  • P(exactly one) = 7/18 × 11/17 + 11/18 × 7/17 = 77/306 + 77/306 = 154/306 = 77/153

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