Combined events
AND and OR
- AND (both happen): multiply the probabilities — for independent events.
- OR (either happens): add the probabilities — for mutually exclusive events.
Practice
For two independent events both happening (AND), you:
P(A and B) = P(A) × P(B) for independent events.
Three pictures
- Sample space diagram: a table of every outcome (e.g. two dice → $36$ cells).
- Venn diagram: sorts outcomes into overlapping sets — read "both" $(A\cap B)$ and "either" $(A\cup B)$.
- Tree diagram: a branch per stage; multiply along branches, then add the paths you want.
Practice
On the 6 by 6 grid of two dice (36 cells), how many cells give a total of 7?
1+6, 2+5, 3+4, 4+3, 5+2, 6+1 — six cells, so P(7) = 6/36 = 1/6.
With and without replacement
- With replacement: chances stay the same. Bag $3$ red, $5$ blue → $\text{P}(\text{red, red}) = \tfrac38 \times \tfrac38 = \tfrac{9}{64}$.
- Without replacement (Extended): one fewer counter next time → $\text{P}(\text{red, red}) = \tfrac38 \times \tfrac27 = \tfrac{3}{28}$.
Practice
Bag of 3 red, 5 blue, two draws WITH replacement. P(red, red) = 9/b. What is b?
(3/8) × (3/8) = 9/64, so b = 64.
Practice
Same bag, WITHOUT replacement. After one red is taken, how many reds remain?
Started with 3 red; one taken leaves 2 red (out of 7 counters).
You've got it
Key idea
- AND → multiply (independent); OR → add (mutually exclusive)
- on a tree: multiply along branches, add the wanted paths
- without replacement, the second probability changes ($\tfrac38 \to \tfrac27$)