| --- |
| source: ncert |
| topic: probability |
| class_level: class_11 |
| chapter: ch16 |
| difficulty: intermediate |
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| |
| 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). |
