Ch 5. & Ch 6

INTRODUCING PROBABILITY

• Context: Introduction to the fundamental concepts of probability.

• Notation: Various symbols (e.g., radius, survival time) are indicative of statistical parameters used in probability.

• Office Hours: Mentioned as 3-4 PM, likely for student consultations.

PROBABILITY DEFINITION

• Probability Experiment: An experiment where individual outcomes cannot be predicted, but the overall results can be anticipated over numerous trials.

  • Example: Tossing a fair coin; individual toss outcomes are uncertain, yet over many tosses, the expectation is approximately 50% heads and 50% tails.

  • Mathematical Representation: For a fair coin, the probability of heads is mathematically expressed as $P(Heads) = rac{1}{2}$ and similarly for tails: $P(Tails) = rac{1}{2}$.

LAW OF LARGE NUMBERS

• Principle: As a probability experiment is repeated indefinitely, the proportion of times a specific event occurs will converge to its actual probability.

SAMPLE SPACE

• Definition: The set of all possible outcomes of a probability experiment is termed as the sample space.

• Examples:

  • Roll of a Die: Sample space $S = {1, 2, 3, 4, 5, 6}$.

  • Coin Toss: Sample space $S = {Heads, Tails}$.

  • Selecting Students: For randomly selecting a student from a list of 10,000, the sample space contains those 10,000 students.

  • Random Sampling of 100 from 10,000: Every distinct group of 100 people from the population is considered in the sample space.

• M&M Example: Sample space $S = {blue, green, red, brown, orange, yellow}$.

EVENTS AND PROBABILITY MODELS

• Event Definition: An event consists of outcomes or a set of possible outcomes and is a subgroup of the sample space.

  • Example Event: Selecting a blue or green M&M.

• Probability Model: A mathematical portrayal of a random phenomenon comprising:

  • Sample space $S$

  • A system for assigning probabilities to events.

• Example Table: Color and probability for M&Ms:

  • Blue: 0.19

  • Green: 0.20

  • Red: 0.15

  • Brown: 0.135

  • Orange: 0.18

  • Yellow: 0.145

CLASSICAL (THEORETICAL) PROBABILITY

• Definition: If an experiment involves $n$ equally likely outcomes, the probability attributed to each outcome is $P = rac{1}{n}$.

• Die Example: Rolling a fair six-sided die with sample space $S = {1, 2, 3, 4, 5, 6}$.

  • Each outcome is equally probable: $P(outcome) = rac{1}{6}$.

ADDITION RULE AND RULE OF COMPLEMENTS

• General Addition Rule: For events A and B,

  • $P(A ext{ or } B) = P(A) + P(B) - P(A ext{ and } B)$.

• Law of Total Probability: The sum of probabilities for all outcomes must equal 1.

• Law of Complements: The probability of an event E not occurring is expressed as $P(not E) = 1 - P(E)$.

EXAMPLES OF PROBABILITY MODELS

• Arranging Children: A couple wanting three children with equal probability for male or female (1/2).

  • Arrangement of children (Boys and Girls):

  • BBB: $P = rac{1}{8}$

  • BBG: $P = rac{1}{8}$

  • BGB: $P = rac{1}{8}$

  • GBB: $P = rac{1}{8}$

  • GGB: $P = rac{1}{8}$

  • GBG: $P = rac{1}{8}$

  • BGG: $P = rac{1}{8}$

  • GGG: $P = rac{1}{8}$

How to Interpret Probability

• Range: Probability $P(E)$ exists between 0 and 1.

  • If $P(E) = 1$: Event E will definitely occur.

  • If $P(E) <br>ightarrow 1$: Very likely to occur.

  • If $P(E) = 0.5$: Even chance of occurrence.

  • If $P(E) <br>ightarrow 0$: Unlikely occurrence.

  • If $P(E) = 0$: Impossible event.

UNUSUAL EVENTS

• Definition: An event with a small probability is termed unusual.

  • Rule of Thumb: Any event with a probability less than 0.05 is labeled as unusual.

• Example: In a college of 5000 students, where 150 are math majors:

  • Probability: $P(Math ext{ Major}) = rac{150}{5000} = 0.03$ (unusual).

THEORETICAL VS. EMPIRICAL PROBABILITY

• Theoretical Probability: Expected outcome based on total possible outcomes.

  • Model: $P = rac{ ext{Number of outcomes in the event}}{ ext{Total possible outcomes}}$.

• Empirical Probability (Experimental Probability): Actual occurrences from repeated experiments.

  • Model: $P = rac{ ext{Number of times the event occurs}}{ ext{Total trials}}$.

  • Example: Number of green marbles: 6 from 10 = 60% theoretical probability, 28 from 50 = 56% empirical probability.

EXAMPLES OF EMPIRICAL METHOD

• Birth Weight Data:

  • Low Birth Weight: 313,752 instances out of 3,791,712 total births:

  • $P(Low ext{ Birth Weight}) <br>ightarrow rac{313752}{3791712} <br>ightarrow 0.0827$.

SAMPLING AS A PROBABILITY EXPERIMENT

• Population Distribution:

  • Families categorized in a town (10,000):

  • Homeowners: 4753

  • Condo Owners: 1478

  • Renters: 912 + 2857 = 3769 total renters.

  • Probability Calculations:

  • $P(Own ext{ a house}) = rac{4753}{10000} = 0.4753$.

  • $P(Rent) = rac{3769}{10000} = 0.3769$.

SECTION 5.2: ADDITION RULE AND RULE OF COMPLEMENTS

• Computational Objectives:

  • Use General Addition Rule to compute probabilities.

  • Use Addition Rule for mutually exclusive events.

  • Employ Rule of Complements for calculations.

PROBABILITY RULES

• Consolidated Probabilistic Framework:

  • Probability of event E: $0 ext{ } ≤ ext{ } P(E) ext{ } ≤ ext{ } 1$.

  • Law of Total Probability: Sum of probabilities for outcomes equals 1.

  • Law of Complements: Probability of the complement event $P(not E) = 1 - P(E)$.

  • General Addition Rule: For events A and B,

  • $P(A ext{ or } B) = P(A) + P(B) – P(A ext{ and } B)$.

EXAMPLES AND APPLICATIONS OF PROBABILITY RULES

• Survey Example: 1000 adults surveyed on educational law preferences and voting intentions, arranging responses into a probability model and deducing the relevant probabilities as detailed in the provided data set.

MUTUALLY EXCLUSIVE EVENTS

• Definition: Events A and B are mutually exclusive if both cannot occur simultaneously.

• Applications: Determining whether specific random selections are mutually exclusive (e.g., rolling certain values on a die).

ADDITION RULE FOR MUTUALLY EXCLUSIVE EVENTS

• If events A and B are mutually exclusive:

  • $P(A ext{ and } B) = 0$.

  • Simplified Addition Rule: $P(A ext{ or } B) = P(A) + P(B)$.

COMPLEMENT OF AN EVENT

• Concept: The event that A does not happen is termed the complement of A, denoted as $A^C$.

• Example: If there is a 60% chance of rain, then 40% is the chance of no rain: $P(not rain) = 1 - P(rain) = 0.40$.

RANDOM VARIABLES

• Definition: A random variable is a numerical representation of outcomes from a probability experiment.

• Types of Random Variables:

  • Discrete Random Variables: Can be distinctly counted (e.g., number of likes, siblings).

  • Continuous Random Variables: Can take on any value in a range (e.g., height, energy consumption).

PROBABILITY DISTRIBUTION

• Discrete Random Variable Distribution: Specifies probabilities for each potential numerical value of the random variable.

• Properties:

  • Each probability lies between 0 and 1 inclusive: $0 ≤ P(x) ≤ 1$.

  • Total probabilities sum to 1: $extstyle ext{∑} P(x) = 1$.

HISTOGRAMS IN PROBABILITY DISTRIBUTION

• Usage: Visual representation of probability distributions using histograms to summarize outcomes (sum of two dice example).

PROBABILITY AND EMPIRICAL EXAMPLES

• Finding Probabilities: Various scenarios (e.g., children per family, number of teenagers texting) computed using the established distributions and methods.

SUMMARY AND APPLICATION OF RULES

• Understanding Probability: Applying core concepts such as addition rules, complements, and exclusive events to practical examples.

• Practical Exercises: Engaging with exercises around classifying events, calculating probabilities and confirming understanding through check-ins and examples.