A probability runs from 0 to 1, and the probabilities of all outcomes total 1.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The and rule
For independent events, P(A and B) = P(A) × P(B).
The or rule
P(A or B) = P(A) + P(B) − P(A and B). Subtracting the overlap stops it being counted twice.
Mutually exclusive
If two events cannot both happen there is no overlap, so the probabilities simply add.
The complement
P(not A) = 1 − P(A). For "at least one", this is almost always the fastest route.
Check independence
Drawing without replacement destroys independence, because the second draw faces a changed set.
Or means add, minus the overlap
P(A or B) = P(A) + P(B) − P(A and B). The subtraction removes outcomes counted twice. Only for mutually exclusive events is the overlap zero and the correction unnecessary.
And means multiply, if independent
P(A and B) = P(A)·P(B) requires independence. Otherwise the second probability must be conditional on the first, and assuming independence without checking is the deeper error.
Mutually exclusive is not independent
Events that cannot both occur are maximally dependent — knowing one happened tells you the other did not. The two terms sound similar and describe nearly opposite situations.
At least one usually means complement
The probability of at least one success is easier as 1 minus the probability of none. Recognising that shortcut removes most of the work from a whole class of problems.
Step 2: Try It Yourself
Tap and try it out.
Red
Blue
Blue has the most. It has 2 more than Red.
Step 3: Watch an Example
One step at a time.
Watch Diego Use the Complement
Diego tosses three fair coins and wants the probability of at least one head.
- Step 1
Counting the ways to get at least one head is fiddly, so he counts the opposite.
Step 4: Your Turn
Practice makes it stick.
The Two Coins
Problem 1 of 2
Two fair coins. Probability both are heads, as a decimal?
The Complement
Problem 2 of 2
Three fair coins. Probability of at least one head, as a decimal?
Combine the Events
1 of 8
P(A) = 0.4, P(B) = 0.5, independent. P(A and B)?
2 of 8
P(A) = 0.3, P(B) = 0.2, mutually exclusive. P(A or B)?
3 of 8
P(A) = 0.6. What is P(not A)?
4 of 8
P(A) = 0.5, P(B) = 0.4, P(A and B) = 0.2. P(A or B)?
5 of 8
Two dice. Probability both show six, to three decimal places?
6 of 8
A bag has 4 red and 6 blue. Probability of red, as a decimal?
7 of 8
Sort each pair by whether the events are independent.
Tap something to move it.
- Empty
- Empty
8 of 8
P(A) = 0.25, P(B) = 0.8, independent. P(A and B)?
Step 5: Quick Check
Show what you know.
Question 1 of 2
P(A) = 0.5, P(B) = 0.3, independent. What is P(A and B)?
Question 2 of 2
Why does the or rule subtract the overlap?
What You Learned
- Multiply for "and" when events are independent.
- Add for "or", subtracting the overlap.
- The complement rule handles "at least one" fastest.