Probability
1. Random Experiment and Sample Space
Random experiment is something you do whose result you cannot predict exactly, but you know all possible results. Like tossing a coin - you know it will be Head or Tail, but not which. Sample space S is set of all possible outcomes. For coin toss S={H,T}. For die throw S={1,2,3,4,5,6}. For two coins S={HH,HT,TH,TT}. Sample space is like menu card - all options listed.
Random experiment: outcome uncertain but all possible outcomes known. Sample space S = set of all possible outcomes. Example coin S={H,T}, die S={1..6}, two coins S={HH,HT,TH,TT}. Outcome = one element of S.
• Random experiment: Result uncertain, but all possible results known
• Sample space S: Set of all possible outcomes
• Example coin: {H,T}, die: {1,2,3,4,5,6}, two coins: {HH,HT,TH,TT}
• Outcome: Single element of S
• Menu card: S shows all options
2. Events - What We Care About
Event is subset of sample space. If sample space is all outcomes, event is some outcomes you care about. Simple event has one outcome, compound has more. Impossible event has no outcome, like getting 7 on die → empty set Φ, probability 0. Sure event is whole sample space S, probability 1. Example die even number event A={2,4,6}. Getting 3 or 5 event B={3,5}.
Event E ⊆ S. Simple event single outcome, compound more than one. Impossible event Φ, P=0. Sure event S, P=1. Example die even A={2,4,6}, odd B={1,3,5}. Event occurs if outcome ∈ E.
• Event E is subset of S, E ⊆ S
• Simple: One outcome {H}, Compound: {2,4,6}
• Impossible: Φ = {} P=0, Sure: S P=1
• Example die even {2,4,6}, odd {1,3,5}
• Event occurs if outcome belongs to E
3. Algebra of Events - Not, Or, And
Complement A' = not A, all outcomes in S not in A. Example A={2,4,6}, A'={1,3,5}. Union A∪B = A or B, at least one happens. Intersection A∩B = A and B, both happen. De Morgan: (A∪B)' = A'∩B', (A∩B)' = A'∪B'. Like 'not (tea or coffee)' = 'no tea and no coffee'.
Complement A' = S - A = not A. Union A∪B = A or B. Intersection A∩B = A and B. Difference A-B = A but not B. De Morgan: (A∪B)'=A'∩B', (A∩B)'=A'∪B'. P(A')=1-P(A).
• Complement A' = not A = S-A, P(A')=1-P(A)
• Union A∪B = A or B (at least one)
• Intersection A∩B = A and B (both)
• De Morgan: (A∪B)'=A'∩B', (A∩B)'=A'∪B'
• Example: A even {2,4,6} A' odd {1,3,5}
4. Mutually Exclusive and Exhaustive
Mutually exclusive means cannot happen together, no common outcome. Example on die, A={1,2} and B={3,4} → A∩B=Φ, so exclusive. If they overlap, not exclusive. Exhaustive means together they cover whole sample space. Example even {2,4,6} and odd {1,3,5} together cover {1..6}=S, so exhaustive. For probability addition, exclusive makes formula simple.
Mutually exclusive: A∩B=Φ, no common outcome, cannot occur together. Exhaustive: A∪B=S, together cover S. Pair can be both exclusive and exhaustive like even and odd on die. More than two events can be pairwise exclusive and exhaustive.
• Mutually exclusive: A∩B=Φ, no overlap, cannot together
• Example: A={1,2} B={3,4} exclusive, A={1,2} B={2,3} not
• Exhaustive: A∪B=S, together cover full S
• Example: Even {2,4,6} + Odd {1,3,5}=S exhaustive
• Both: Even and odd are exclusive and exhaustive
5. Axiomatic Approach - Three Rules
Kolmogorov gave 3 axioms that all probabilities must follow. Axiom 1 Non-negativity: P(A)≥0 probability never negative. Axiom 2 Normalization: P(S)=1 total probability 1 =100%. Axiom 3 Additivity: If A and B are disjoint (exclusive), then P(A∪B)=P(A)+P(B). From these, all other rules come like P(Φ)=0, P(A')=1-P(A), 0≤P(A)≤1.
Three axioms: 1) P(A)≥0 for any event A. 2) P(S)=1. 3) If A∩B=Φ then P(A∪B)=P(A)+P(B). Consequences: P(Φ)=0, 0≤P(A)≤1, P(A')=1-P(A), P(A∪B)=P(A)+P(B)-P(A∩B).
• Axiom 1: P(A)≥0 non-negative
• Axiom 2: P(S)=1 whole sample space 100%
• Axiom 3: If A∩B=Φ then P(A∪B)=P(A)+P(B) additivity
• Results: P(Φ)=0, 0≤P≤1, P(A')=1-P(A)
• Foundation: All probability from these 3
6. Probability of an Event
For equally likely outcomes, P(A)= Number of favourable outcomes / Total outcomes = n(A)/n(S). Example die even: n(A)=3, n(S)=6, P=3/6=1/2. Properties: 0≤P(A)≤1, P(Φ)=0, P(S)=1, P(A')=1-P(A). If you know P(A), you know P(not A). Example probability of not getting even =1-1/2=1/2 which is odd.
Equally likely: P(A)=n(A)/n(S). 0≤P≤1. P(S)=1, P(Φ)=0, P(A')=1-P(A). n(A)=favourable, n(S)=total. If not equally likely, use axioms or given probabilities.
• Formula (equally likely): P(A)=n(A)/n(S)=favourable/total
• Range: 0≤P(A)≤1
• P(S)=1 sure, P(Φ)=0 impossible
• Complement: P(A')=1-P(A)
• Example: Die even n(A)=3 n(S)=6 P=1/2, not even 1-1/2=1/2
7. Addition Theorem - P(A or B)
What is probability A or B happens? General: P(A∪B)=P(A)+P(B)-P(A∩B). Subtract overlap because counted twice. If A and B are mutually exclusive, P(A∩B)=0, so P(A∪B)=P(A)+P(B). Example die: A even {2,4,6} P=1/2, B {3,4,5} P=1/2, A∩B={4} P=1/6, so P(A∪B)=1/2+1/2-1/6=5/6. Check: A∪B={2,3,4,5,6} 5/6 correct.
Addition: P(A∪B)=P(A)+P(B)-P(A∩B). If mutually exclusive P(A∩B)=0 then P(A∪B)=P(A)+P(B). For three events P(A∪B∪C)=P(A)+P(B)+P(C)-P(A∩B)-P(B∩C)-P(C∩A)+P(A∩B∩C). Use Venn.
• General: P(A∪B)=P(A)+P(B)-P(A∩B) subtract overlap
• Exclusive: If A∩B=Φ then P(A∪B)=P(A)+P(B)
• Example: Even 1/2 + {3,4,5} 1/2 - {4} 1/6 =5/6
• Three events: Add singles - pairs + triple
• Use: A or B means union
8. Real Life Usage - Probability Everywhere
Probability is everyday. Weather forecast 70% rain = probability 0.7. Insurance uses probability to set premium. Games: Ludo dice 6 probability 1/6. Lottery, cricket toss. Medical: Probability test positive if disease. Quality control: Probability defective item. Business: Probability demand high. Even your phone face unlock uses probability to match face. Understanding mutually exclusive helps avoid double counting.
Real: Weather forecast %, insurance premium based on P(event), games dice cards, lottery, toss, medical diagnosis P(disease|test), quality control defective, business demand, face unlock matching probability, search engine, AI.
• Weather: 70% rain = P=0.7
• Games: Dice P(6)=1/6, cards, toss, Ludo
• Insurance: Premium based on P(accident)
• Medical: P(disease|positive test)
• Quality: P(defective) in factory
• Tech: Face unlock probability match, AI predictions
• Sample Space S: All possible outcomes, Event E⊆S
• Types: Simple one outcome, compound many, Φ impossible P0, S sure P1
• Algebra: A' not A, A∪B or, A∩B and, De Morgan (A∪B)'=A'∩B'
• Exclusive Exhaustive: A∩B=Φ exclusive, A∪B=S exhaustive
• Axioms: P≥0, P(S)=1, if exclusive P(A∪B)=P(A)+P(B)
• Probability: P(A)=n(A)/n(S), 0≤P≤1, P(A')=1-P(A)
• Addition: P(A∪B)=P(A)+P(B)-P(A∩B), if exclusive add directly
• Real: Weather %, games, insurance, medical, quality, face unlock