Probability

Chapter 4: Probability

Introduction

  • Understanding the basics of probability is essential for statistical inference methods.

  • Example: Mice survival under different dietary restrictions illustrates probability of observing differences in outcomes.

  • Key statistical inference question: What is the probability of observed differences assuming no actual effects?

Basics of Probability

Interpreting Probability

  • Probability experiments: events whose outcomes cannot be predicted with certainty (e.g., tossing a coin, rolling dice).

  • Long-run characteristics: Over many trials, the distribution of outcomes approaches a stable pattern.

  • Frequentist Interpretation: Probability as a long-term proportion of times an outcome occurs.

Sample Spaces and Sample Points

  • Sample Space (S): The set of all possible outcomes of an experiment.

    • Example 1: Rolling a die: S = {1, 2, 3, 4, 5, 6}.

    • Example 2: Drawing a card from a deck: S = {A♣, 2♣, 3♣, ..., Q♠, K♠}.

  • The sample points must be mutually exclusive and collectively exhaustive.

Events

  • An event is a subset of the sample space represented typically by capital letters (A, B, C).

  • Example: Defining events based on outcomes from rolling dice (E, F, G).

  • Useful to visualize events using Venn diagrams.

Rules of Probability

  • Probability Rules:

    • For any event A: 0 ≤ P(A) ≤ 1.

    • For any sample space S: P(S) = 1.

Intersection of Events

  • The intersection of two events A and B, written as A ∩ B, is when both events occur.

Mutually Exclusive Events

  • Events with no shared outcomes: P(A ∩ B) = 0; called disjoint events.

Union of Events and Addition Rule

  • The union of two events A and B, written as A ∪ B, occurs when at least one of the events occurs.

    • Addition Rule: P(A ∪ B) = P(A) + P(B) - P(A ∩ B).

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

Complementary Events

  • Complement of event A, denoted A^c, is the event that A does not occur.

  • Probability relationships:

    • P(A) + P(A^c) = 1

    • P(A^c) = 1 - P(A).

Conditional Probability

  • Conditional Probability (P(A|B)): The probability of event A occurring given that event B has occurred.

  • Formula: P(A|B) = P(A ∩ B) / P(B).

  • Example calculations through various scenarios.

Independence of Events

  • Events A and B are independent if P(A ∩ B) = P(A)P(B).

  • An intuitive understanding of independence through betting scenarios.

Multiplication Rule

  • Multiplication Rule for two events: P(A ∩ B) = P(A)P(B|A) = P(B)P(A|B).

  • If A and B are independent: P(A ∩ B) = P(A)P(B).

Examples and Applications

  • Various examples illustrate key concepts in probability, conditional probabilities, examples using real-life applications, and further exploration through tree diagrams.

  • Combinations and permutations are useful for calculating possible outcomes in random sampling and decision scenarios.

Summary

  • Key definitions and formulas are reiterated:

    • Sample Space, Events, Probabilities, Conditional Events, Independence, Multiplication Rule, and Counting Formulas.

    • Each event's relevance in statistical inference and various practical applications in decision-making, health statistics, lottery odds, etc.

Key Formulas in Probability

  1. Probability of an Event: [ P(A) \text{ for any event } A: 0 \leq P(A) \leq 1 ] [ P(S) = 1 \text{ (for the sample space S)} ]

  2. Addition Rule: [ P(A \cup B) = P(A) + P(B) - P(A \cap B) ]

    • Special case for mutually exclusive events: [ P(A \cup B) = P(A) + P(B) ]

  3. Complementary Events: [ P(A^c) = 1 - P(A) ] [ P(A) + P(A^c) = 1 ]

  4. Conditional Probability: [ P(A|B) = \frac{P(A \cap B)}{P(B)} ]

  5. Independence of Events: [ P(A \cap B) = P(A)P(B) ]

  6. Multiplication Rule: [ P(A \cap B) = P(A)P(B|A) = P(B)P(A|B) ]

    • If A and B are independent: [ P(A \cap B) = P(A)P(B) ]

  1. Basic Probability: What is the probability of rolling a 4 on a standard six-sided die?

  2. Sample Space: If you flip a coin and roll a die at the same time, what is the sample space for this experiment?

  3. Events: Define two events, A (rolling an even number) and B (rolling a number greater than 4) when rolling a die. List the outcomes for each event and determine if they are mutually exclusive.

  4. Addition Rule: Using events from the previous question, calculate P(A ∪ B).

  5. Complementary Events: If P(A) = 0.3, what is P(A^c)?

  6. Conditional Probability: If we have a deck of cards, what is the probability of drawing a Queen given that you have already drawn a heart?

  7. Independence: Let A be the event of raining today and B be the event of having class scheduled. Are A and B independent if it's known that class is scheduled regardless of the weather? Explain your reasoning.

  8. Multiplication Rule: What is the probability of flipping heads on a coin and rolling a 5 on a die?

  9. Real-Life Application: A bag contains 3 red balls and 2 blue balls. If you draw one ball at random, what is the probability that it is blue? What is the probability of not drawing a blue ball?

  10. Combinations: How many ways can you choose 2 balls from the 5 balls in the previous question?

Here are the answers to the practice problems:

  1. Basic Probability: The probability of rolling a 4 on a standard six-sided die is ( P(4) = \frac{1}{6} ).

  2. Sample Space: If you flip a coin and roll a die at the same time, the sample space is: ( S = {(Heads, 1), (Heads, 2), (Heads, 3), (Heads, 4), (Heads, 5), (Heads, 6), (Tails, 1), (Tails, 2), (Tails, 3), (Tails, 4), (Tails, 5), (Tails, 6)} ) which has a total of 12 outcomes.

  3. Events:

    • Event A (rolling an even number): ( {2, 4, 6} )

    • Event B (rolling a number greater than 4): ( {5, 6} )

    • These events are not mutually exclusive since they share the outcome '6'.

  4. Addition Rule: Using events from the previous question, ( P(A \cup B) = P(A) + P(B) - P(A \cap B) ). Here, ( P(A) = \frac{3}{6} = \frac{1}{2} ), ( P(B) = \frac{2}{6} = \frac{1}{3} ), and ( P(A \cap B) = P(6) = \frac{1}{6} ).

    Thus, [ P(A \cup B) = \frac{1}{2} + \frac{1}{3} - \frac{1}{6} = \frac{3}{6} + \frac{2}{6} - \frac{1}{6} = \frac{4}{6} = \frac{2}{3} ]

  5. Complementary Events: If ( P(A) = 0.3 ), then ( P(A^c) = 1 - P(A) = 1 - 0.3 = 0.7 ).

  6. Conditional Probability: The probability of drawing a Queen given that you have already drawn a heart is ( P(Q|H) = \frac{P(Q \cap H)}{P(H)} = \frac{1/52}{1/4} = \frac{1}{13} ) since there's 1 Queen of hearts and 13 total hearts.

  7. Independence: Events A (raining today) and B (class scheduled) are independent because the occurrence of rain doesn’t affect the scheduling of class. ( P(A | B) eq P(A) ) under usual circumstances, thus they are considered independent.

  8. Multiplication Rule: The probability of flipping heads on a coin and rolling a 5 on a die is ( P(H) \times P(5) = \frac{1}{2} \times \frac{1}{6} = \frac{1}{12} ).

  9. Real-Life Application: The probability of drawing one blue ball from a bag containing 3 red balls and 2 blue balls is ( P(Blue) = \frac{2}{5} ). The probability of not drawing a blue ball is ( P(Not Blue) = 1 - P(Blue) = 1 - \frac{2}{5} = \frac{3}{5} ).

  10. Combinations: The number of ways to choose 2 balls from 5 is given by ( \binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = 10 ).