Unit 12: Probability

1. Random Experiments and Sample Space
2. Types of Events
3. Axiomatic Approach to Probability
4. Probability of 'And', 'Or', and 'Not' Events
5. Exam Focus Enhancements

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, A_n are exhaustive if their union is the sample space (A₁ ∪ A₂ ∪ \dots ∪ A_n = 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:

1. 0 ≤ P(A) ≤ 1
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

Exam Tips

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).

Common Mistakes

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.

Frequently Asked Questions

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).