Probability — Class 11 Maharashtra HSC Mathematics revision notes
A 4-page visual revision sheet for Probability, written for Class 11 Maharashtra HSC Mathematics. Formulas, diagrams and the traps that cost marks, laid out to be read the night before rather than the term before. Free, and it opens inside CapCompass.
Page 1 as PDF
Probability in one look
events are just sets, and every probability rule is set theory wearing a disguise
Basic terminology
- a random experiment has more than one known outcome, but the actual result can't be predicted
- the sample space S (or Ω) is the set of ALL possible outcomes. n(S) counts them
- an event is any subset of the sample space; a favourable outcome belongs to that subset
- elementary event: one outcome. certain event: all of S. impossible event: the empty set φ
Algebra of events
- Acup B: outcomes belonging to at least one of A, B
- Acap B: outcomes belonging to both A and B
- A and B are exhaustive if Acup B=S
- A and B are mutually exclusive (disjoint) if Acap B=φ, no outcome in common
- A, B are complementary if BOTH mutually exclusive AND exhaustive: Acup B=S, Acap B=φ
Reading combinations of events
| Operation | Interpretation |
|---|---|
| A' | not A |
| Acup B | at least one of A, B |
| Acap B | both A and B |
| (A'cap B)cup(Acap B') | exactly one of A, B |
| (Acup B)' | neither A nor B |
Union and intersection from a definition
Q. S is the positive integers up to 50. A is the subset divisible by 6, B the subset divisible by 9. Find Acup B and Acap B
- A=｛6, 12, 18, 24, 30, 36, 42, 48｝, B=｛9, 18, 27, 36, 45｝
- Acup B collects every element of either set
- Acap B keeps only what both share: 18 and 36, the multiples of lcm(6,9)=18
Answer: Acup B=｛6,9,12,18,24,27,30,36,42,45,48｝, Acap B=｛18, 36｝
Classical probability
- definition, all outcomes equally likely
P(A)=n(A)/n(S)
Elementary properties
- Complement: P(A')=1-P(A)
- Bounds: 0≤ P(A)≤ 1
- Impossible, certain: P(φ)=0, P(S)=1
- Mutually exclusive: P(Acup B)=P(A)+P(B)
- Subset: Asubset B⇒ P(A)≤ P(B)
- Set difference: P(Acap B')=P(A)-P(Acap B)
Page 1 of 4. The full sheet, all 4 pages, opens free in the app.
What this Probability sheet covers
- Basic terminology
- Algebra of events
- Reading combinations of events
- Union and intersection from a definition
- Classical probability
- definition, all outcomes equally likely
- Elementary properties
- Complement
- Bounds
- Impossible, certain
- Mutually exclusive
- Subset
- Set difference
- Probability by direct counting
- Probability from an arrangement count
- Addition theorem
- for any two events A, B
- Addition theorem on two dice
- Conditional probability
- probability of A given B has occurred
- Conditional probability, reduced sample space
- Multiplication theorem and independence
- multiplication theorem, always true
- A, B independent means
- so for independent events
- Without replacement breaks independence
- Bayes' theorem
- E_1,...,E_n mutually exclusive, exhaustive, P(E_i)≠0
- Bayes' theorem on a repair record
- Odds
- m favourable of n total, so P(A)=m/n
- equivalently
- odds against A
- Odds from a die
- Traps that cost marks
- Tricks that save time
How CapCompass uses this sheet
These notes are not a standalone PDF. CapCompass plans Mathematics chapter by chapter against your real exam dates and free hours, and when the plan reaches Probability the sheet opens inside the study block itself — at the page covering the sub-topic that block is on, with the timer still running. Revision arrives at the moment the plan says revise, instead of having to be gone looking for.