Addition Rule of Probability (Mutually Exclusive Events)
For mutually exclusive events, P(A or B) = P(A) + P(B). If the events can overlap, subtract P(A and B) so the overlap is not counted twice.
THE IDEA
The Addition Rule of Probability (The 'OR' Rule)
This principle covers calculating the probability of one event or another occurring when the two events are mutually exclusive (they cannot happen at the same time).
The Calculation: P(A or B) = P(A) + P(B)
Example from the text (Jelly Beans): Consider picking a jelly bean from a jar. Imagine the probabilities are:
- Red: 0.5
- Yellow: 0.4
- Green: 0.1
Question: What is the probability of picking a red or a yellow? Answer: P(Red) + P(Yellow) = 0.5 + 0.4 = 0.9
Question: What is the probability of picking a yellow or a green? Answer: P(Yellow) + P(Green) = 0.4 + 0.1 = 0.5
Campbell emphasizes reasoning through why the OR rule adds these probabilities, rather than merely memorizing the arithmetic.
Key Limitation: This simple addition rule only applies when the events are mutually exclusive (no overlap). If events can co-occur, the formula changes to the General Addition Rule: P(A or B) = P(A) + P(B) - P(A and B).
When to use it
Estimating technical-fault or budget-shortfall risk: add the probabilities only if both cannot occur in the defined outcome; otherwise subtract their overlap.
Calculating the probability that a customer selects Product A OR Product B from a specific two-item display, given they purchase exactly one.
Calculating whether one specified shot is a free throw OR a layup, when these are mutually exclusive shot categories.
FROM THEORY TO EVERYDAY LIFE
In practice
Jelly Bean Probability
A jar contains jelly beans with specific probabilities: Red (0.5), Yellow (0.4), Green (0.1). The probability of picking a red OR a yellow bean is 0.5 + 0.4 = 0.9. The probability of picking a yellow OR a green bean is 0.4 + 0.1 = 0.5. The colors are mutually exclusive for a single draw.
Simple Deck of Cards
What is the probability of drawing a heart OR a diamond from a standard deck of cards? The events 'drawing a heart' (P=13/52=0.25) and 'drawing a diamond' (P=13/52=0.25) are mutually exclusive. The probability is 0.25 + 0.25 = 0.5.
A MOMENT TO REFLECT
Put it to work.
Can you think of a decision you face where you implicitly add probabilities for an 'OR' scenario? Is that addition mathematically valid, or do the events have overlap?
Why did Campbell claim people don't remember probability rules by practicing them? What makes the addition rule for 'OR' counterintuitive despite its arithmetic simplicity?
How would the probability of picking a red or a yellow jelly bean change if a bean could be both red and yellow?
Often confused with
CONTINUE THE CONVERSATION
Source texts
- The Thinkers Toolkit by Morgan D Jones