Probability measures how likely an event is to occur. It is a number between 0 and 1, where 0 means the event is impossible and 1 means it is certain. In Class 11, we study probability using set theory and the axiomatic approach.
- Key Definitions:
- Experiment: An action with observable outcomes (e.g., rolling a die).
- Sample Space (S): The set of all possible outcomes. |S| = total number of outcomes.
- Event (E): A subset of the sample space.
- Probability of Event E: P(E) = n(E)/n(S)
- Types of Events:
- Mutually Exclusive Events: Events that cannot occur together. P(A ∩ B) = 0.
- Exhaustive Events: Events whose union is the entire sample space.
- Equally Likely Events: Each outcome is equally probable.
- Complementary Event: P(A') = 1 - P(A)
- 1.Axiomatic Definition:
- 2.For any event E in sample space S:
- 3.0 ≤ P(E) ≤ 1
- 4.P(S) = 1
- 5.For mutually exclusive events: P(A ∪ B) = P(A) + P(B)
Addition Theorem:
P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
This is the most important formula in this chapter. For mutually exclusive events, P(A ∩ B) = 0.
- Important Results:
- P(A') = 1 - P(A)
- P(A ∩ B') = P(A) - P(A ∩ B)
- P(A' ∩ B') = 1 - P(A ∪ B) [De Morgan's Law applied to probability]
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Worked Examples
A fair coin is tossed twice. What is the sample space?
S = {HH, HT, TH, TT}. |S| = 4.
Two dice are rolled. Find P(sum = 7).
Favourable outcomes: (1,6),(2,5),(3,4),(4,3),(5,2),(6,1) — 6 outcomes.
P = 6/36 = 1/6
A card is drawn from a standard deck of 52 cards. Find P(king or heart).
P(King) = 4/52, P(Heart) = 13/52, P(King of Hearts) = 1/52.
P(King or Heart) = 4/52 + 13/52 - 1/52 = 16/52 = 4/13
If P(A) = 0.4, P(B) = 0.5, P(A ∪ B) = 0.7, find P(A ∩ B).
P(A ∩ B) = P(A) + P(B) - P(A ∪ B) = 0.4 + 0.5 - 0.7 = 0.2
A bag has 3 red and 5 blue balls. One is drawn at random. Find P(red).
Total = 8. P(red) = 3/8.
Events A and B are mutually exclusive with P(A) = 0.3 and P(B) = 0.4. Find P(A ∪ B).
P(A ∪ B) = 0.3 + 0.4 = 0.7 (since A and B are mutually exclusive, P(A ∩ B) = 0)
A number is chosen from 1 to 20. What is the probability that it is divisible by 3 or 5?
Divisible by 3: {3,6,9,12,15,18} — 6 numbers. Divisible by 5: {5,10,15,20} — 4 numbers.
Divisible by both (15): 1 number. P = (6+4-1)/20 = 9/20
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Common mistakes
- P(A ∪ B) = P(A) + P(B) is ONLY valid when A and B are mutually exclusive. In general, you must subtract P(A ∩ B).
- The number of elements in the sample space must account for all outcomes — do not overlook any.
- Probability cannot exceed 1 or be negative. If your answer is outside [0, 1], recheck your work.
Summary
Probability is the ratio of favourable outcomes to total outcomes. The addition theorem P(A ∪ B) = P(A) + P(B) - P(A ∩ B) and the complementary rule P(A') = 1 - P(A) are the core tools in this chapter.