Unit 12: Probability
Contents
1. Random Experiments and Sample Space
A Random Experiment is an experiment whose all possible outcomes are known in advance, but the exact outcome of any specific performance cannot be predicted.
- Outcomes: The possible results of a random experiment.
- Sample Space (S): The set of all possible outcomes.
Example: Tossing two coins, S = \{HH, HT, TH, TT\}. - Sample Point: Each element of the sample space.
2. Types of Events
An Event is any subset of a sample space. We say an event A has occurred if the outcome of the experiment is in set A.
- Impossible Event: An empty set (φ), which contains no outcomes.
- Sure Event: The entire sample space (S).
- Simple Event: An event containing only one sample point.
- Compound Event: An event containing more than one sample point.
Mutually Exclusive and Exhaustive Events
- Mutually Exclusive Events: Two events A and B are mutually exclusive if they cannot happen at the same time (A ∩ B = φ).
- Exhaustive Events: Events A1, A2, \dots, An are exhaustive if their union is the sample space (A1 ∪ A2 ∪ \dots ∪ An = S).
3. Axiomatic Approach to Probability
The axiomatic approach is a modern way of defining probability using three fundamental rules (axioms) for any event A in sample space S:
2. P(S) = 1
3. If A and B are mutually exclusive, P(A ∪ B) = P(A) + P(B)
4. Probability of 'And', 'Or', and 'Not' Events
We use set theory operations to calculate the probability of complex events.
- Probability of 'Not' A (A'): The probability that event A does not occur. P(A') = 1 - P(A)
- Probability of 'A or B' (A ∪ B): Addition Theorem of Probability. P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
- Probability of 'A and B' (A ∩ B): The probability that both events occur simultaneously.
5. Exam Focus Enhancements
- The Sum Rule: Always remember that the sum of probabilities of all simple events in a sample space is exactly 1.
- Mutually Exclusive Shortcut: If a question says events are "mutually exclusive," immediately set P(A ∩ B) = 0 in the addition theorem.
- At Least One: The phrase "probability of at least one event occurring" usually means P(A ∪ B).
- P > 1: If your calculated probability is greater than 1 or negative, there is an error in your calculation. Probability must be between 0 and 1.
- Confusing Mutually Exclusive vs Independent: Mutually exclusive means they can't happen together (P(A ∩ B) = 0). Independent is a different concept (where P(A ∩ B) = P(A) · P(B)).
- Sample Space Count: Forgetting to list all outcomes. For example, tossing 3 coins has 23 = 8 outcomes, not 6.
Q: What is the probability of an Impossible Event?
A: The probability is always 0.
Q: What are 'Equally Likely' outcomes?
A: Outcomes are equally likely if none of them has a preference to occur over others (like a fair die or coin).
Q: How do I find P(A ∩ B')?
A: This is the probability of "A but not B," calculated as P(A) - P(A ∩ B).