Probability Rules

Mutually Exclusive and Independent Events

Learning Objectives

Understand what it means for events to be mutually exclusive or independent.

Mutually Exclusive Events

  • Two events that cannot occur at the same time are considered to be mutually exclusive
  • The probability of both events occurring at once will equal zero

Independent Events

  • Spotting independent events is a little bit more involved
  • Remember, P(A|B) = the probability of A occurring given that B has already occurred.
  • What happens if B has no effect on A? Then P(A|B) = P(A)
  • That also means that P(A and B) = P(A|B)×P(B) =P(A)×P(B)

Testing for Mutually Exclusive Events (Example)

Example 19.1

Problem Setup: Let us have two possible events, 1 and 2, that can either occur or not occur. See below:

Event 1 Not Event 1 Totals
Event 2 0 P(not E1 and E2) P(E2)
Not Event 2 P(E1 and not E2) P(not E1 and not E2) P(not E2)
Totals P(E1) P(not E1) 1

Question: Which events are mutually exclusive based on the above contingency table?

Solutions:

  • Look for any ‘zeros’ in the table.
  • In this case, P(E1 and E2) = 0
  • So, events 1 and 2 are mutually exclusive

Testing for Independence (Example)

  • We can test for any events where P(A|B) = P(A)
  • Or we can test if P(A and B) =P(A)×P(B)
  • If this is true, then the events (we call them ‘A’ and ‘B’ for simplicity here) are independent
  • See the next examples of this testing.

Example 19.2

Problem Setup: Let us take a simple example of coffee and tea drinkers. Let’s use the following notation:

  • P(T) = probability of drinking tea
  • P(C) = probability of drinking coffee

See the contingency table for coffee and tea drinkers below:

Drinks Coffee Doesn’t Drink Coffee Totals
Drinks Tea 0.6 0.075 0.675
Doesn’t Drink Tea 0.2 0.125 0.325
Totals 0.8 0.2 1

Question: Are drinking coffee and tea independent events?

Solutions: Let’s start by writing out the values we will use from the above table:

  • P(Drinks Coffee) = P(C) = 0.8
  • P(Drinks Tea) = P(T) = 0.675
  • P(Drinks Coffee and Drinks Tea) = P(C and T) = 0.6

Let’s test, is P(C and T) = P(C)×P(T)?

  • P(C)×P(T) = 0.8×0.675 = 0.54
  • P(C and T) = 0.6 ≠ 0.54

Conclusion: Because P(C and T) ≠ P(C)×P(T), coffee and tea drinking are not independent events. Ie: whether someone drinks coffee has an effect on whether or not they choose to drink tea.

Key Takeaways (EXERCISE)

Key Takeaways: Mutually Exclusive and Independent Events

 

Your Own Notes (EXERCISE)

  • Are there any notes you want to take from this section? Is there anything you’d like to copy and paste below?
  • These notes are for you only (they will not be stored anywhere)
  • Make sure to download them at the end to use as a reference

License

Icon for the Creative Commons Attribution-NonCommercial 4.0 International License

An Introduction to Business Statistics for Analytics (1st Edition) Copyright © 2024 by Amy Goldlist; Charles Chan; Leslie Major; Michael Johnson is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License, except where otherwise noted.

Share This Book