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Probability Topics: Venn Diagrams (optional)

Module by: Susan Dean, Dr. Barbara Illowsky

Summary: This module introduces Venn diagrams as a method for solving some probability problems. This module is included in the Elementary Statistics textbook/collection as an optional lesson.

A Venn diagram is a picture that represents the outcomes of an experiment. It generally consists of a box together with circles or ovals. The circles or ovals represent events.

Example 1

Suppose an experiment has the outcomes 1, 2, 3, ... , 12 where each outcome has an equal chance of occurring. Let event A = {1, 2, 3, 4, 5, 6}A = {1, 2, 3, 4, 5, 6} and event A = {6, 7, 8, 9}A = {6, 7, 8, 9}. Then A AND B = {6}A AND B = {6} and A OR B = {1, 2, 3, 4, 5, 6, 7, 8, 9}A OR B = {1, 2, 3, 4, 5, 6, 7, 8, 9}. The Venn diagram is as follows:

A Venn diagram. An oval representing set A contains the values 1, 2, 3, 4, 5, and 6. An oval representing set B also contains the 6, along with 7, 8, and 9. The values 10, 11, and 12 are present but not contained in either set.

Example 2

Flip 2 fair coins. Let AA = tails on the first coin. Let BB = tails on the second coin. Then A = {TT, TH} A={TT,TH} and B = {TT, HT}B={TT,HT}. Therefore, A AND B = {TT}A AND B = {TT}. A OR B = {TH, TT, HT}A OR B = {TH,TT,HT}.

The sample space when you flip two fair coins is S = {HH, HT, TH, TT}S ={HH,HT, TH,TT}. The outcome HHHH is in neither AA nor BB. The Venn diagram is as follows:

Venn diagram with set A containing Tails + Heads and Tails + Tails, and set B containing Tails + Tails and Head + Tails. Head + Heads is contained in neither set, and set A and set B share Tails + Tails.

Example 3

Forty percent of the students at a local college belong to a club and 50% work part time. Five percent of the students work part time and belong to a club. Draw a Venn diagram showing the relationships. Let CC = student belongs to a club and PT PT = student works part time.

Venn diagram with one set containing students in clubs and students in clubs and working part-time and another set containing C/PT and students working part-time. Both sets share C/PT.

  • The probability that a students belongs to a club is P(C) = 0.40P(C)= 0.40.
  • The probability that a student works part time is P(PT) = 0.50P(PT)=0.50.
  • The probability that a student belongs to a club AND works part time is P(C AND PT) = 0.05P(C AND PT)= 0.05.
  • The probability that a student belongs to a club given that the student works part time is:
    P(C|PT)  =  P(C AND PT) P(PT)  =  0.05 0.50  =  0.1 P(C|PT) =  P(C AND PT) P(PT)  =  0.05 0.50  = 0.1 (1)
  • The probability that a student belongs to a club OR works part time is:
    P(C OR PT) = P(C) + P(PT) - P(C AND PT) = 0.40 + 0.50 - 0.05 = 0.85 P(C OR PT) = P(C) + P(PT) - P(C AND PT) = 0.40 + 0.50 - 0.05 = 0.85 (2)

Glossary

Venn Diagram:
The useful visual representation of a sample space and events in the form of circles or ovals showing their intersections.

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