Lesson 11 of 11 · 15 min
Chapter review
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NCERT probability revised in detail
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Probability at JEE Main level
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Must-know facts
17 facts
- 1An event is any subset of the sample space; a sample space of n outcomes has 2ⁿ events.
- 2E occurs when the outcome ω ∈ E.
- 3Impossible event φ; sure event S.
- 4Simple event: one sample point; n outcomes give exactly n simple events. Compound event: more than one point.
- 5not A = A′ = S − A; A or B = A ∪ B; A and B = A ∩ B; A but not B = A − B = A ∩ B′.
- 6Mutually exclusive: A ∩ B = φ. Exhaustive: the union is S.
- 7Mutually exclusive and exhaustive: exactly one of the events occurs every time.
- 8Axioms: P(E) ≥ 0, P(S) = 1, P(E ∪ F) = P(E) + P(F) for disjoint E, F.
- 9P(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
Treating the 11 possible sums of two dice as equally likely.
Adding P(A) and P(B) for events that overlap.
Thinking mutually exclusive means exhaustive, or the other way round.
Accepting a probability table because every entry lies between 0 and 1.
Using m/n on a weighted die or unfair coin.
Counting 'at least one' case by case and missing a case.
Writing P(neither A nor B) = 1 − P(A) − P(B) for overlapping events.
Using permutations for a committee or a hand of cards.
Taking HT and TH as one outcome when two coins are tossed.
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.