Addition rule
Why subtract intersection?: If events overlap, we count intersection twice.
Subtract once to correct.
Worked example
P(A)=0.4, P(B)=0.3, P(both)=0.1.
Find P(A or B).
Solution
- P(A or B)=0.4+0.3-0.1
- P(A or B)=0.6
Final answer
0.6 (subtract 0.1 to avoid double-counting).
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Multiplication rule and independence
Independent events: Events A and B independent if P(B|A)=P(B).
Then: P(A and B)=P(A)×P(B).
Worked example
A train is late with probability 0.2. On any day, independently, it rains with probability 0.3.
Find the probability that the train is late and it rains.
Solution
- P(late) = 0.2, P(rain) = 0.3. Independent: one does not change the other.
- P(late and rain) = 0.2 × 0.3 = 0.06
Final answer
0.06. Independent, so multiply.
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Conditional probability
Meaning: Probability of B given A happened.
Restricts sample space to outcomes where A occurs.
Worked example
Cards: 5 red, 3 black.
Pick one.
If red, prob of second red?
(no replacement)
Solution
- Given first is red: 4 red left, 7 total left
- P(second red | first red)=4/7
Final answer
4/7. Sample space shrinks after first pick.
Show it: mutually exclusive or independent?
| Claim | Test | Write |
|---|---|---|
| Mutually exclusive | P(A ∩ B) = 0 | Both can happen (the overlap is not empty), so NOT mutually exclusive. |
| Independent | P(A ∩ B) = P(A) × P(B) | Work out both sides and compare them. |
Mutually exclusive?
In a class of 28, 9 students play the piano, 12 play football and 4 do both. Are "plays the piano" and "plays football" mutually exclusive? Justify your answer.
Solution
- Mutually exclusive means they cannot happen together, so P(both) would be 0.
- 4 students do both.
Final answer
No: 4 students play both, so P(P ∩ F) = 4/28 ≠ 0. The events are not mutually exclusive.
Independent?
P(A) = 0.4, P(B) = 0.5 and P(A ∩ B) = 0.2. Are A and B independent?
Solution
- Multiply the single probabilities.
- Compare with the given P(A ∩ B).
Final answer
Yes: P(A) × P(B) = 0.2 = P(A ∩ B), so A and B are independent.
Exam tip: "Justify" = the number that shows it AND the conclusion. "No" alone scores 0.