P(A or B) = P(A) + P(B) − P(A and B). The subtraction stops the overlap being counted twice.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The multiplication rule
P(A and B) = P(A) × P(B|A). Only when the events are independent does that become P(A) × P(B).
Testing independence
A and B are independent when P(B|A) = P(B) — knowing A happened changes nothing.
Not the same as mutually exclusive
Mutually exclusive events cannot both happen, which makes them strongly dependent, not independent.
The addition rule
P(A or B) = P(A) + P(B) − P(A and B). The subtraction removes the double count of the overlap. Only for mutually exclusive events is the overlap zero and the subtraction unnecessary.
The multiplication rule
P(A and B) = P(A)·P(B | A). Only when the events are independent does this reduce to multiplying the unconditional probabilities. Assuming independence without checking is the commonest probability error.
Conditional probability changes the denominator
P(A | B) restricts attention to the cases where B happened. That changes what you are dividing by, which is why conditional and unconditional probabilities can differ enormously.
Mutually exclusive is not independent
Mutually exclusive events cannot both happen, so knowing one occurred tells you the other did not — which makes them maximally dependent. The two terms are routinely confused and mean nearly opposite things.
Step 2: Try It Yourself
Tap and try it out.
favourable and the rest
3
7
all outcomes
?
The hatched bar is the piece the story is asking for.
Step 3: Watch an Example
One step at a time.
Watch Rosa Avoid Double Counting
P(A) = 0.5, P(B) = 0.4, P(A and B) = 0.2. Rosa finds P(A or B).
- Step 1
Adding gives 0.9, but the overlap has been counted twice.
Step 4: Your Turn
Practice makes it stick.
Or
Problem 1 of 2
P(A) = 0.3, P(B) = 0.5, P(A and B) = 0.1. What is P(A or B)?
And
Problem 2 of 2
Independent events with P(A) = 0.5 and P(B) = 0.6. What is P(A and B)?
Which Rule
1 of 8
Independent, P(A) = 0.2, P(B) = 0.5. P(A and B)?
2 of 8
Mutually exclusive, P(A) = 0.3, P(B) = 0.4. P(A or B)?
3 of 8
Mutually exclusive events. What is P(A and B)?
4 of 8
P(A) = 0.6. What is P(not A)?
5 of 8
P(B|A) = P(B). Independent? 1 for yes, 0 for no.
6 of 8
Are mutually exclusive events independent? 1 for yes, 0 for no.
7 of 8
Two fair coins. P(both heads)?
8 of 8
Select every statement that is always true.
Step 5: Quick Check
Show what you know.
Question 1 of 1
Independent, P(A) = 0.4, P(B) = 0.5. P(A and B)?
What You Learned
- The addition rule subtracts the overlap so it is not counted twice.
- The multiplication rule uses a conditional probability unless the events are independent.
- Mutually exclusive is the opposite of independent, not a version of it.