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source: ncert
topic: probability
class_level: class_11
chapter: ch16
difficulty: intermediate
---
Class 11 | Ch16: Probability
RANDOM EXPERIMENT: outcome cannot be predicted with certainty.
SAMPLE SPACE S: set of all possible outcomes.
EVENT: subset of sample space. Simple event: one outcome. Compound: multiple outcomes.
ALGEBRA OF EVENTS:
Complement of A: A'=S-A. A∪A'=S. A∩A'=phi.
Mutually exclusive: A∩B=phi (cannot occur together).
Exhaustive: A∪B=S (together cover all outcomes).
AXIOMATIC APPROACH TO PROBABILITY:
P(S)=1. 0<=P(A)<=1. P(A∪B)=P(A)+P(B) if A,B mutually exclusive.
ADDITION THEOREM:
P(A∪B)=P(A)+P(B)-P(A∩B).
P(A∪B∪C)=P(A)+P(B)+P(C)-P(A∩B)-P(B∩C)-P(A∩C)+P(A∩B∩C).
CONDITIONAL PROBABILITY:
P(A|B)=P(A∩B)/P(B). (Probability of A given B has occurred.)
INDEPENDENT EVENTS:
A and B independent if P(A∩B)=P(A)×P(B).
Equivalent: P(A|B)=P(A). (Occurrence of B doesn't affect A.)
MULTIPLICATION THEOREM:
P(A∩B)=P(A)×P(B|A)=P(B)×P(A|B).
SOLVED EXAMPLES:
Example 1: P(A)=0.4, P(B)=0.5, P(A∩B)=0.2.
P(A∪B)=0.4+0.5-0.2=0.7.
P(A|B)=P(A∩B)/P(B)=0.2/0.5=0.4.
(Note: P(A|B)=P(A)=0.4, so A and B are INDEPENDENT.)
Example 2: Card drawn. P(red or king).
P(red)=26/52. P(king)=4/52. P(red king)=2/52.
P(red or king)=26/52+4/52-2/52=28/52=7/13.
Example 3: Two dice. Find P(sum>9).
Favourable: (4,6),(5,5),(5,6),(6,4),(6,5),(6,6)=6 outcomes.
P=6/36=1/6.
Example 4: P(A)=1/4, P(B)=1/3. A,B independent. Find P(A or B).
P(A∩B)=1/4×1/3=1/12.
P(A∪B)=1/4+1/3-1/12=3/12+4/12-1/12=6/12=1/2.
COMMON MISTAKES:
Mutually exclusive: P(A∩B)=0. Independent: P(A∩B)=P(A)×P(B). DIFFERENT concepts.
Mutually exclusive events with nonzero probabilities are NOT independent.
P(A|B) is NOT the same as P(B|A). |