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Chapter 5: Probability

5.1 Probability Rules

Learning Objectives

• Apply the rules of probabilities.

• Compute and interpret probabilities using the empirical method.

• Compute and interpret probabilities using the classical method.

• Use simulation to obtain data based on probabilities.

• Recognize and interpret subjective probabilities.

Introduction to Probability

Definition: Probability is a quantitative measure of the likelihood that a random phenomenon or chance behavior will occur. It describes the long-term proportion of outcomes over repeated trials where short-term outcomes may vary significantly.

Example: To visualize the concept of probability, one can simulate flipping a coin 100 times and observe the proportion of heads versus tails. For example, if 55 heads and 45 tails were observed, the empirical probability of getting heads would be 0.55.

Key Concepts:

1. Probability Experiments: Yield random short-term outcomes but exhibit long-term predictability.

   • Example: Rolling a die multiple times will yield random results each time (1 through 6), but the long-term probability of each face showing up remains consistent (1/6).

2. Sample Space (S): The collection of all possible outcomes of a probability experiment, which can include one or more outcomes.

   • Example: For a coin flip, the sample space is S = {Heads, Tails}.

3. Events: Any collections of outcomes are categorized as events, which can be simple (a single outcome, such as a specific number when rolling a die) or complex (comprising multiple outcomes).

   • Example of Simple Event: Rolling a 3 on a die.

   • Example of Complex Event: Rolling an even number on a die (which includes outcomes 2, 4, and 6).

4. Simple Events: Nomenclature includes denoting simple events as e_i (e.g., e_1, e_2) while events are generally captured by capital letters like E.

Law of Large Numbers

As the number of repetitions of a probability experiment increases, the observed frequencies of outcomes converge to the theoretical probabilities. This law is a foundational principle in probability, highlighting the importance of larger sample sizes for more accurate predictions.

• Example: If you flip a fair coin 10 times, you may not get exactly 5 heads and 5 tails. However, if you flip it 1000 times, the proportion of heads should be very close to 0.5.

5.1.1 Apply the Rules of Probabilities

Probability Range:

The probability of an event E is constrained within the range: 0 ≤ P(E) ≤ 1.

Total Probability:

The sum of probabilities of all individual outcomes in a sample space must equal 1: [ P(e_1) + P(e_2) + ... + P(e_n) = 1 ]

Probability Model:

A comprehensive framework listing all possible outcomes of a probability experiment along with the probability of each outcome, ensuring compliance with the rules mentioned.

Characterization of Events:

• Impossible event: P(E) = 0

  • Example: The probability of rolling a 7 on a standard six-sided die.

• Certain event: P(E) = 1

  • Example: The probability of rolling a number less than 7 on a standard six-sided die.

• Unusual event: Typically has a low probability, indicating occurrences outside of regular expectations.

  • Example: The probability of getting a 12 when rolling two six-sided dice (which is only possible with a roll of 6 on both).

5.1.2 Compute Using the Empirical Method

Empirical Probability Approximation:

The empirical probability can be derived and approximated using the formula: [ P(E) ≈ \frac{\text{Number of times E occurs}}{\text{Total number of experiments}} ] Example - Pass the Pigs Game: A class of 52 students rolled pig figurines 3,939 times, meticulously recording the outcomes to analyze the frequencies and build a probability model from these results.

Building a Probability Model:

Utilize the collected results to construct a model reflecting the probability of different landing outcomes, setting reasonable expectations like predicting approximately 329 outcomes of “side with dot” in 1,000 throws.

5.1.3 Compute Using the Classical Method

Equally Likely Outcomes:

When outcomes of an event are equally likely to occur, use the classical method, defined as: [ P(E) = \frac{\text{Number of ways E can occur}}{\text{Total number of outcomes}} ] Example - M&M Candy: In a diverse bag containing M&Ms of different colors, calculate the probability for each color based on their respective quantities (e.g., P(yellow) = 6/30; P(blue) = 2/30).

• If there are 6 yellow, 10 red, 8 green, and 6 blue candies in a bag (30 candies total), then each color's probability could be calculated as the respective number of each color divided by total, e.g., P(red) = 10/30 = 1/3.

5.1.4 Simulation

Simulation Practice:

Utilize various simulation tools to replicate actions such as rolling dice, allowing learners to compare empirical results against classical probabilities, solidifying understanding of probability concepts.

• Example: Simulating the rolling of two dice 10,000 times to see the distribution of sums. Theoretical probability can be compared to the empirical results obtained from the simulation, illustrating any discrepancies and fostering deeper understanding of random events.

5.1.5 Subjective Probabilities

Definition: Subjective probabilities are derived from personal judgment or belief rather than from direct empirical data or frequency counts.Example: Economic predictions about future recessions reflect an economist’s subjective probability assessment, focusing on risk evaluation rather than strict statistical outcomes.

• Example Case Study: Hal Stern examined horse race betting trends, showcasing how betting amounts indicate subjective probabilities—a method that does not rely on empirical measurements but rather reflects the opinions and beliefs of gamblers about horse performances.

5.2 The Addition Rule and Complements

Learning Objectives

• Use the Addition Rule for disjoint events.

• Utilize the General Addition Rule.

• Compute probability of an event using the Complement Rule.

5.2.1 Use the Addition Rule for Disjoint Events

Disjoint Events: Events that cannot occur simultaneously are termed mutually exclusive.

Addition rule for disjoint events:

[ P(E \text{ or } F) = P(E) + P(F) ]

Venn Diagrams:

These visual tools are employed to graphically represent events and their probabilities, facilitating a clearer understanding of intersections and unions of events.Example - Housing Units: Validating a probability model for housing units may require graphical representation through Venn diagrams to show the relationship of events such as "house sold" and "house rented."

5.2.2 General Addition Rule Application:

When dealing with events that are not disjoint, the General Addition Rule applies: [ P(E \text{ or } F) = P(E) + P(F) - P(E \text{ and } F) ] Example - Dice Roll: Demonstrates the application of the rule in scenarios where outcomes from both events can occur simultaneously, such as rolling two dice where you might want the probability of rolling a 3 on die 1 or a 5 on die 2. Here you'll need to subtract the overlap where both conditions are fulfilled, should such scenarios exist.

5.2.3 Complement Rule

Definition: The complement of an event E, noted E^C, encapsulates all outcomes not included in E. [ P(E^C) = 1 - P(E) ]Example - Travel Time: Analyzing the probability that a randomly selected resident has a commuting time less than 90 minutes could utilize complementary probability: If the probability of selecting someone with a commuting time of 90 minutes or more is 0.3, then the probability of selecting one with a commuting time less than 90 minutes would be 1 - 0.3 = 0.7.

5.3 Independence and the Multiplication Rule

Learning Objectives

• Identify independent events.

• Use the Multiplication Rule for independent events to compute probabilities.

5.3.1 Identify Independent Events

Definition: Events E and F are considered independent if the occurrence of one does not influence the likelihood of the other occurring.Examples: Comparing outcomes when drawing cards from a deck and simultaneously rolling a die can illustrate independence in actions. For instance, drawing an Ace from a shuffled deck (without replacement) does not affect the probability of rolling a “4” on the die.

5.3.2 Multiplication Rule for Independent Events

Rule Application:[ P(E \text{ and } F) = P(E) \times P(F) ]Example: If the probability of rolling a 5 on a die is 1/6 and the probability of drawing an Ace from a deck of cards (with 52 cards) is 4/52 (or 1/13), then the probability of both events occurring (rolling a 5 and drawing an Ace) would be [ P(5 \text{ and } Ace) = \frac{1}{6} \times \frac{1}{13} = \frac{1}{78} ].

5.3.3 At-Least Probabilities

Calculate probabilities for events categorized as “at least one” occurrence, employing understanding from the multiplication rule.

• Example: In a survey assessing if at least one out of four students who bought a textbook scored above average, if the probability of each student scoring above average is 0.3, then the probability that at least one student scores above average can be computed using the complement rule instead of calculating individual scenarios directly.

5.4 Conditional Probability and General Multiplication Rule

Learning Objectives

• Compute conditional probabilities.

• Use the General Multiplication Rule for events that may not be independent.

5.4.1 Conditional Probability

Definition: The probability P(F|E) represents the likelihood that event F occurs given that event E has happened.

• Example: If the probability of a person being a smoker is 0.2 and the probability of a smoker having lung cancer is 0.25, then the conditional probability that a randomly chosen person has lung cancer given that they are a smoker would be 0.25. This highlights how conditional situations adjust the understanding of probabilities.

5.5 Counting Techniques

Learning Objectives

• Solve counting problems effectively using the Multiplication Rule.

• Gain understanding of permutations and combinations and applications of both in probability.

5.5.1 Multiplication Rule of Counting Application:

When a task involves independent choices in a sequence, the total number of ways to select outcomes can be calculated as the product of the number of choices available at each stage.

• Example: If there are 3 choices to make for breakfast and 4 choices for drinks, the total combinations of breakfast and drinks would be 3 x 4 = 12.

5.5.2 Permutations and Combinations

Definitions:

• Permutations: Represent ordered arrangements of objects, where the sequence matters.

  • Example: Arranging 3 books in 3 slots (3! = 6 combinations).

• Combinations: Represent selections where the order of things does not matter, highlighting the diverse applications of counting in probability scenarios.

  • Example: Choosing 2 fruits from a selection of 5 fruits. The number of ways to do this is represented by ( C(5,2) = \frac{5!}{2!(5-2)!} = 10 ).

5.6 Putting It Together

Learning Objectives

• Determine the most appropriate probability rule or counting technique applicable in various decision-making scenarios.

5.7 Bayes’s Rule

Learning Objectives

• Employ total probabilities and Bayes's rule to update probabilities based on new data, forming a critical part of decision-making processes in statistics and probability theory.

• Example: In diagnosing diseases, if a test has a known false positive rate, Bayes's theorem can be applied to assess the probability of a patient truly having the disease after a positive test result considering the test accuracy and overall disease prevalence.