Basic Probability Notes

Basic Probability

Random Experiment

  • Definition: An activity that produces an outcome, known as a random experiment.

  • Characteristics: The outcome cannot be predicted with certainty.

  • Example: Tossing a coin, where heads or tails cannot be determined beforehand.

Sample Space

  • Definition: The set of all possible outcomes of an experiment.

  • Denotation: Usually represented by 'S'.

  • Example for tossing 2 coins:

    • S = {TT, HH, HT, TH}

    • n(S) = 4 (the number of outcomes)

  • Example for rolling a die:

    • S = {1, 2, 3, 4, 5, 6}

    • n(S) = 6

Event

  • Definition: An event is a subset of a sample space and is denoted by 'E'.

  • Example: Getting an even number when rolling a die.

    • Let S be the sample space; E = {2, 4, 6}

    • n(E) = 3 (number of favorable outcomes).

Types of Events

  • Complement of an Event: If E is an event, then its complement (denoted as E') is the event that E does not occur.

  • Property: n(E) + n(E') = n(S)

  • Simple Event: An event that consists of a single outcome (e.g., tossing a coin and getting heads).

  • Compound Event: An event that consists of two or more outcomes.

  • Sure Event: If the event space is equal to the sample space (the event will definitely occur).

  • Impossible Event: An event with an empty event space (e.g., rolling a die and getting a number greater than 6).

Equally Likely Events

  • Events are said to be equally likely if there is no reason to favor one over the other; each event has an equal chance of occurring.

Probability

  • Definition: A numerical measure of the likelihood of the occurrence of an event, denoted P(E).

  • Formula: For an event E,

  • P(E) = n(E) / n(S), provided that all outcomes are equally likely.

  • Properties of Probability:

  • 0 ≤ P(E) ≤ 1

  • P(E) + P(E') = 1

  • If A ⊆ B, then P(A) ≤ P(B)

Conditional Probability

  • Definition: The probability of an event A occurring given that event B has occurred, denoted as P(A|B).

  • Formula:

  • P(A|B) = P(A ∩ B) / P(B), provided P(B) > 0

Examples of Probability Calculations

  1. Tossing three coins:

  • Total number of outcomes: 2^3 = 8

  • Event of getting all heads: E = {HHH} → P(E) = 1/8

  • Event of getting at least one head: E = {HHH, HHT, HTH, HTT, THH, THT, TTH} → P(E) = 7/8.

  1. Rolling Two Dice:

  • Total outcomes = 36 (6 sides on Die 1 x 6 sides on Die 2).

  • Calculate the probability of getting a sum of 10 or 11.

Bayes' Theorem

  • Provides a way to update the probability estimate for an event based on new evidence.

  • Formula:

  • P(B|A) = [P(A|B) * P(B)] / P(A)

  • Useful for determining the conditional probability when multiple events are involved.

Mutually Exclusive Events

  • Events A and B are mutually exclusive if they cannot occur simultaneously.

  • Rule for Addition:

  • P(A ∪ B) = P(A) + P(B) for mutually exclusive events.

Conclusion

  • Understanding basic concepts of probability includes recognizing events, calculating probabilities, and applying rules such as Bayes' theorem to assess outcomes.