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)
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
-
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: Mutually Exclusive and Independent Events
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