For independent events, P(A and B) = P(A) × P(B). Two fair coins both landing heads is 0.5 × 0.5.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
What independence means
Two events are independent when one happening does not change the chance of the other.
The addition rule
P(A or B) = P(A) + P(B) − P(A and B). The subtraction removes the overlap counted twice.
Mutually exclusive events
If two events cannot both happen there is no overlap, so the probabilities simply add.
Two different ideas
Mutually exclusive is not the same as independent. In fact exclusive events are strongly dependent: one occurring makes the other impossible.
With and without replacement
Drawing without replacement destroys independence, because the second draw faces a changed bag.
Or means add, minus the overlap
P(A or B) = P(A) + P(B) − P(A and B). Forgetting the subtraction double-counts the outcomes in both, which is the standard error whenever the events are not mutually exclusive.
And means multiply, if independent
P(A and B) = P(A)·P(B) only when the events are independent. Otherwise the second probability must be conditional on the first. Assuming independence without checking is the deeper error.
What independence means
Two events are independent when knowing one occurred does not change the probability of the other. Successive coin flips are independent; drawing cards without replacement is not.
Mutually exclusive is not independent
If two events cannot both happen, knowing one occurred tells you the other did not — which is maximum dependence. The two terms sound similar and describe nearly opposite situations.
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 Ines Handle an Overlap
One card is drawn. Ines wants the probability it is a heart or a king.
- Step 1
There are 13 hearts in 52 cards, so P(heart) = 13/52.
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 Overlap
Problem 2 of 2
P(A) = 0.5, P(B) = 0.4, P(A and B) = 0.2. What is P(A or B)?
Combine Them
1 of 8
P(A) = 0.3, P(B) = 0.5, independent. P(A and B)?
2 of 8
P(A) = 0.2, P(B) = 0.3, mutually exclusive. P(A or B)?
3 of 8
Two dice. Probability both show six, as a decimal to three places?
4 of 8
P(A) = 0.6, P(B) = 0.5, P(A and B) = 0.3. P(A or B)?
5 of 8
Are mutually exclusive events independent? 1 for yes, 0 for no.
6 of 8
Three fair coins. Probability all heads, as a decimal to three places?
7 of 8
Sort each pair by whether the events are independent.
Tap something to move it.
- Empty
- Empty
8 of 8
P(A) = 0.4, P(B) = 0.25, independent. P(A and B)?
Step 5: Quick Check
Show what you know.
Question 1 of 2
P(A) = 0.5, P(B) = 0.2, independent. What is P(A and B)?
Question 2 of 2
Why does the addition rule subtract P(A and B)?
What You Learned
- For independent events, multiply to get "and".
- For "or", add and subtract the overlap.
- Mutually exclusive is not the same as independent.