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AI HL · 13.1

Probability

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13.1.2

Venn diagrams for probability

13.1.2

Venn diagrams for probability

Target: I can fill a two-set Venn diagram, starting with the overlap and working outwards. (+3 more)

By the end of this section

Key points

Drawing the Venn diagram

  • The rectangle is the whole group U. Each circle is one set. Where circles overlap is "both".
  • Put the overlap in first. A total for a set includes its overlap, so subtract the overlap to get "only".
  • The four regions must add to n(U). The region outside both circles is "neither".
  • If "both" and "neither" are given, find "only A" or "only B" by subtracting from the total.

Reading probabilities

  • \(P(A)=\frac{n(A)}{n(U)}\): add all the regions inside circle A, including the overlap.
  • "A but not B" is only the part of A outside the overlap.
  • "Neither" is the region outside both circles: \(P(\text{neither})=1-P(A\cup B)\).
  • If the diagram shows probabilities, not counts, the four regions add to 1.

Explore: two events in a Venn diagram

Type P(A), P(B) and P(A ∩ B). The four regions fill in. Choose what to shade and read the working. The circles are not drawn to scale.

Shade

P(A ∪ B)
P(A′)
P(A | B)
P(B | A)
P(A) × P(B)

Type three probabilities to fill the diagram.

  1. Start with P(A) = 0.5 and P(B) = 0.4. Change only P(A ∩ B) until the events are independent. What value does it need, and how do you know?
  2. Choose "Mutually exclusive". Look at P(A | B). Why is it 0? Are the events independent?
  3. Compare P(A | B) with P(B | A). When are they the same, and why is it usually not so?
  4. Try a P(A ∩ B) bigger than P(A). What does the sentence say, and why can it never happen?

Worked examples

Problems like the ones in class. Pick one, try it, then open the working.
Fill the diagram, then read a probability

In a group of 40 students, 24 play tennis and 18 play squash. 9 play both sports. A student is chosen at random. Find the probability that the student plays tennis but not squash. Give your answer as a decimal.

Working
  1. Intersection first

    Both sports: 9

    Start in the middle and work outwards.

  2. Only tennis, only squash

    Only tennis = 24 − 9 = 15. Only squash = 18 − 9 = 9

    Take the overlap away from each set total.

  3. 15997TS
    Neither

    40 − 15 − 9 − 9 = 7

    The four regions must add to the total.

  4. Probability

    \(P=\frac{15}{40}=\frac{3}{8}\) = 0.375

    Only-tennis region over the total.

Video walkthrough coming
Probability of neither

Of 70 people at a café, 38 ordered pizza and 31 ordered pasta. 17 ordered both. One person is chosen at random. Find the probability that the person ordered neither. Give your answer to 3 significant figures.

Working
  1. Intersection

    Both: 17

    The 17 who ordered both go in the overlap.

  2. Only pizza, only pasta

    38 − 17 = 21. 31 − 17 = 14

    Subtract the overlap from each total.

  3. 21141718PS
    Neither

    70 − (21 + 17 + 14) = 18

    Everyone outside both circles.

  4. Probability

    \(P(\text{neither})=\frac{18}{70}=\frac{9}{35}\) = 0.257

    Give 3 significant figures.

Video walkthrough coming
Both and neither given

In a year group of 120 students, 30 study both Art and Music, 22 study neither, and 41 study Art only. A student is chosen at random. Find the probability that the student studies Music. Give your answer as a decimal.

Working
  1. Fill what you are given

    Both 30, neither 22, Art only 41

    Put all three in the diagram.

  2. 41273022AM
    Music only

    120 − 30 − 22 − 41 = 27

    The four regions add to the total, so the missing one is found by subtraction.

  3. All of Music

    27 + 30 = 57

    The whole Music circle is Music only plus the overlap.

  4. Probability

    \(P(M)=\frac{57}{120}=\frac{19}{40}\) = 0.475

    A decimal is exact here.

Video walkthrough coming