Detailed Notes on Probability
14.1 Event
Probability is fundamentally about events, which are subsets of a sample space. The sample space encompasses all possible outcomes of a random experiment. For example, when tossing a coin twice, the sample space (S) is {HH, HT, TH, TT}.
- Events represent specific outcomes of interest.
- A subset E of S is considered an event:
- Example: E = {HT, TH} refers to the event of getting exactly one head.
14.1.1 Occurrence of an Event
An event occurs when the outcome of a random experiment belongs to the subset defining that event.
- For an experiment like throwing a die, if the event E is defined as “a number less than 4,” then outcomes 1, 2, or 3 mean E has occurred. If any outcome is above 3 (4, 5, 6), E has not occurred.
14.1.2 Types of Events
- Impossible Events: The empty set (φ), where no outcome occurs.
- Example: Rolling a die to get a multiple of 7 (impossible).
- Sure Events: The complete sample space (S), which always occurs.
- Example: Any die roll results in {1, 2, 3, 4, 5, 6}.
- Simple Events: Those with just one outcome in the sample space.
- Example: Tossing a coin and getting heads {H}.
- Compound Events: Events with multiple outcomes.
- Example: Getting at least one head in three tosses.
14.1.3 Algebra of Events
Combining Events Using Set Operations:
- Complementary Event (A′): All outcomes in S that are not part of event A.
- Union of Events (A ∪ B): Outcomes that are in A, in B, or in both.
- Intersection of Events (A ∩ B): Outcomes common to both A and B.
- Example with both A and B as outcomes from rolling a dice: If A = {even numbers} and B = {greater than 3}.
14.1.4 Mutually Exclusive Events
- Definitions: A and B are mutually exclusive if they cannot occur simultaneously (A ∩ B = φ).
- Example: Getting an odd number and an even number in one die throw.
14.1.5 Exhaustive Events
- Events are exhaustive if their union equals the entire sample space (S).
- Example: If A = {1, 2}, B = {3, 4} cover all outcomes {1, 2, 3, 4}.
14.2 Axiomatic Approach to Probability
The axiomatic approach to probability consists of defining probabilities based on a set of axioms.
- Key Axioms:
- Non-negativity P(E) ≥ 0
- Total Probability P(S) = 1
- Additivity for mutually exclusive events: P(E ∪ F) = P(E) + P(F)
- Outcome Probabilities: For any event E, P(E) sums over all elementary events in E.
14.2.1 Probability of an Event
For an event occurring within an experiment, if the total outcomes are n(S), the probability of an event A is calculated as:
P(A) = n(A)/n(S)
14.2.2 Equally Likely Outcomes
When all outcomes of a sample space are equally likely, the probability of an event A is the ratio of the number of favorable outcomes to the total outcomes.
14.2.3 Combined Probability of Events A and B
- For any two events A and B, the combined probability can be represented as:
P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
- This adjustment accounts for double counting in cases where A and B share common outcomes.
14.2.4 Probability of Event 'Not A'
The probability of the complement of A (denoted A′) is given by:
P(A′) = 1 - P(A)
Exercises Summary
The chapter includes exercises to reinforce the understanding of probability concepts by calculating probabilities based on defined events and summarizing outcomes from experiments.
Conclusion
Through events, set operations, and axiomatic definitions, we quantify uncertainty in various contexts. Understanding these foundational concepts prepares individuals for more complex statistical analysis and interpretations.