Lesson 11 of 11 · 15 min
Chapter review
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Must-know facts
17 facts
- 01An event is any subset of the sample space; a sample space of n outcomes has 2ⁿ events.
- 02E occurs when the outcome ω ∈ E.
- 03Impossible event φ; sure event S.
- 04Simple event: one sample point; n outcomes give exactly n simple events. Compound event: more than one point.
- 05not A = A′ = S − A; A or B = A ∪ B; A and B = A ∩ B; A but not B = A − B = A ∩ B′.
- 06Mutually exclusive: A ∩ B = φ. Exhaustive: the union is S.
- 07Mutually exclusive and exhaustive: exactly one of the events occurs every time.
- 08Axioms: P(E) ≥ 0, P(S) = 1, P(E ∪ F) = P(E) + P(F) for disjoint E, F.
- 09P(φ) = 0 follows from the axioms.
- 10Valid assignment: every P(ωᵢ) in [0, 1] and the total is exactly 1.
- 11P(A) = sum of P(ωᵢ) over the outcomes in A.
- 12Equally likely outcomes: P(E) = n(E)/n(S).
- 13P(A ∪ B) = P(A) − P(A ∩ B) + P(B): the overlap is removed once.
- 14Three events: add the singles, remove the pairs, add back P(A ∩ B ∩ C).
- 15P(not A) = 1 − P(A).
- 16Only one of two: P(A) + P(B) − 2P(A ∩ B).
- 17NCERT results: Example 7 gives 0.87, 0.98, 0.11; Example 10 gives 1/7735, 9/1547, 46/7735; Example 12 gives 1/60 and 1/10.
Common traps
Where marks are lost
Formulas
10 to know
Complement
A′ = S − A, P(A′) = 1 − P(A)
A and A′ are mutually exclusive and exhaustive.
Difference
A − B = A ∩ B′
The event 'A but not B'.
Axioms
P(E) ≥ 0, P(S) = 1, P(E ∪ F) = P(E) + P(F) if E ∩ F = φ
Hence P(φ) = 0.
Event from outcomes
P(A) = Σ P(ωᵢ), ωᵢ ∈ A
Each 0 ≤ P(ωᵢ) ≤ 1 and all P(ωᵢ) add to 1.
Equally likely outcomes
P(E) = n(E)/n(S)
Each outcome has probability 1/n(S).
Addition rule
P(A ∪ B) = P(A) − P(A ∩ B) + P(B)
Usually written P(A) + P(B) − P(A ∩ B); the overlap is removed once.
Mutually exclusive events
P(A ∪ B) = P(A) + P(B)
Because P(A ∩ B) = 0.
Three events
P(A ∪ B ∪ C) = S₁ − S₂ + S₃
S₁ = sum of P of singles, S₂ = sum of P of the three pairwise intersections, S₃ = P(A ∩ B ∩ C).
Neither A nor B
P(A′ ∩ B′) = 1 − P(A ∪ B)
De Morgan: A′ ∩ B′ = (A ∪ B)′.
Exactly one of A, B
P(A) + P(B) − 2 P(A ∩ B)
P(A ∩ B′) + P(A′ ∩ B).
Key terms
12 terms
- Random experiment
- An experiment with more than one possible outcome, where the outcome cannot be predicted in advance.
- Sample space
- The set S of all possible outcomes of a random experiment.
- Event
- Any subset of the sample space.
- Impossible event
- The empty set φ; no outcome makes it happen.
- Sure event
- The whole sample space S; every outcome makes it happen.
- Simple event
- An event with exactly one sample point, also called an elementary event.
- Compound event
- An event with more than one sample point.
- Complementary event
- The event 'not A', A′ = S − A, made of the outcomes outside A.
- Mutually exclusive events
- Events that cannot occur together; their intersection is φ.
- Exhaustive events
- Events whose union is S, so at least one of them always occurs.
- Equally likely outcomes
- Outcomes that all have the same probability.
- Axiomatic approach
- Defining probability as a function on events that obeys three rules: non-negative, P(S) = 1, and additive for disjoint events.