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).