Lecture 3

• Study notes for probability are outlined below

What is Probability?

• Probability measures the likelihood of an event occurring under specified conditions.

History

• Probability's formal study began with gambling analysis.
• 10 BC - 54 AD: Emperor Claudius wrote "On the Art of Dice."
• 1654: Chevalier de Mere introduced 'The Problem of Dice'.
• De Mere's favored bet was getting at least one 6 in four dice rolls, which he often won.
  • Single die roll gave 1/6 chance of rolling a 6.
  • The chances for four rolls were calculated as 4 x 1/6 = 0.6667 or 2/3.
  • De Mere also bet on rolling a double-six with two dice in 24 rolls, and he often lost.
• De Mere consulted Pascal to understand the problem, leading to the theory of probability.

Random Processes

• Random processes involve knowing potential outcomes without knowing the specific outcome.
  • Examples are coint tosses, stock fluctuations, or shuffle play.
• Randomness depends on how one defines the process.
  • For example, predicting personal coffee consumption involves free will, while a researcher might correlate it with sleep, dopamine, and academic workload as a random process.

Representing Probability

• Random experiment: Observing the outcome of a single chance event.
  • For example, noting the outcome of a coin toss is a random experiment.
• Random variable: This is the measurement of a random experiment.
  • Can take on a value that describes the outcome (e.g., X = 10).
• Elementary outcomes: Possible results of a random experiment (e.g., heads or tails).
• Sample space: the set of all possible elementary outcomes.
  • Denoted as {H, T}
• Probability: This is the numerical weight describing the likelihood of an outcome's occurrence.
  • The probability of heads or tails in a fair coin toss is 0.5, or P(H) = P(T) = 0.5.

Interpreting Probability

• Event probability reflects how often the event would occur in infinite random experiments within the sample space.
  • The long-term frequency of an event represents the frequentist view.
• The Bayesian view includes the frequentist view plus a person's subjective certainty about an event.
  • Different people may have varying certainty or viewpoints for the same event
  • Examples include the probability that human activity influences global warming.

Random Experiment #1

• A coin flip with a fair coin serves as a random experiment.
• The probability of heads, P(H), is being determined.
• The coin flip has two possible outcomes: heads and tails
• The sample space is {H, T}.
• The probability is calculated as the number of outcomes meeting a condition divided by the number of equally likely outcomes.
  • 1 / 2 = 0.5 = 50%

Mutually Exclusive Events

• "Disjoint" events cannot occur simultaneously.
  • A coin flip cannot result in both heads and tails.
  • One cannot pass and fail a course within the same term.

Non Disjoint Events

• Non-disjoint events can occur together.
  • Getting an A in multiple courses in the same term
  • Flipping two coins could result in both heads and both tails.

Loaded Die

• The probabilities of elementary events cannot exceed 1 (100%), but don't have to be equal for a loaded die.
  • For example, P(1)=.25
  • The other events do not need to be equal

Comparing Probabilities

• P(event) = 0 (or 0%) means the event is impossible and probabilities can't be negative
• P(event) = 1 (or 100%) signifies an absolutely certain event.
• P(event) = 0.5 (or 50%) means the event is equally likely to happen or not happen.
• P(event) = .25 (or 25%) signifies the event is less likely to happen than not.
• P(event) = .75 (or 75%) indicates the event that is more likely to happen than not.

Data Representation

• Pie charts: Show relative proportions of data per category
  • The sum of proportions equals 1 (100%)
• Stacked bar charts: Bar divisions show the relative proportions of data per response type.
  • Often used with survey/poll data.
  • They can be "diverging," aligned to a vertical baseline.

Events as Sets and Logical Operations

• Rolling a pair of fair, 6-sided dice yields 36 outcomes with equal probability.
• Combining events using "Logical Operations" is useful.
• Three logical operators are employed:
  • AND
  • OR
  • NOT
• Given {Event X, Event Y} new evenets can be made
  • Event X AND Event Y occurs if both events occur.
  • Event X OR Event Y occurs if either Event X or Event Y (or both) occurs.
  • NOT Event X occurs if Event X does not occur.
• Addition Rule
  • P(Event 1 OR Event 2) = P(Event 1) + P(Event 2) - P(Event 1 AND Event 2)
  • For disjoint events (where P(Event 1 AND Event 2)=0)
  • P(Event 1 OR Event 2) = P(Event 1) + P(Event 2)

Complimentary Events

• Complimentary Events: Cover or completes the rest of the sample space from what was predicted/occurred
  • The complimentary event ot a coin flip or heads is coinf flip of NOT heads
  • The complimentary event to a die roll of 3 is Die roll of 'NOT 3' with a sample space = {1, 2, 3, 4, 5, 6}

Subtraction Rule

• P(NOT Event 1) = 1 - P(Event 1)
  • P(rolling a 3) = 1/6
  • P(NOT rolling a 3) = 5/6

Probability Distributions

• Listing all possible events with their probabilities is key
• Rules
  • Events are disjoint.
  • Each event’s probability falls between 0 and 1
  • Probabilities must sum to 1

Contingency Table

• A contingency table displays categorical variables in terms of frequencies or relative frequency (%).

Marginal Probability

• Defined as the probability of a single event occurring, regardless of other events.
  • For example, determining the likelihood that a participant did NOT relapse involves considering the total number of participants who did not relapse, without specifying treatment.

Joint Probability

• It determines the likelihood that a participant took desipramine AND relapsed, which represents the intersection of outcome and treatment. P(Desipramine AND Relapse) = 10/72 = 0.1389.139

Independence

• Two variables are independent if Event X's probability, given Event Y, equals Event X's probability by itself Given that the coin 1st toss was heads, what is the probability that coin toss 2 is heads?
• If events are not independent, then we need to think about "conditional probability"
  • For rain, if it is sunny and the sky is cloudless: chance rain is low
  • If it is cloudy, grey, and warmer than 3 degrees: chance of rain is higher
  • Presence of one event makes the presence of a second more/less likely

Conditional Probability

• Conditional probability of the outcome of interest (Event X), given the presence of another condition (Event Y)
  • P(X|Y) = P(X and Y) / P(Y)

Multiplication Rule

• For two outcomes or events, X and Y:
  • P(X and Y) = P(X|Y) x P(Y)
  • This formula is a slightly rearranged conditional probability equation, useful for thinking of X as the outcome of interest and Y as a related condition.

Independence and Conditional Probability

• The hypothetical probability distribution includes gender identity and faculty
• If P(X|Y) = P(X), event X and Y are independent.
  • Knowing Y tells us nothing about X.
  • Mathematically, X and Y are independent if and only if P(X and Y) = P(X) x P(Y).
  • The joint probability is the product of the two marginal probabilities.

Probability and Distributions:

• What does this mean for sampling?
  • Consider the normal distribution
  • What is the probability of randomly drawing a score that falls within 1 standard deviation of the mean?
  • Within 2 SDs?
  • Why is it that the mean of a representative same converges on the true population mean?

Explore the fundamentals of probability, including its historical roots and applications. Starting from gambling analysis, delve into the contributions of figures like Emperor Claudius and Chevalier de Mere, whose challenges led to the development of probability theory. Understand random processes and their reliance on potential outcomes.