Introduction to Frequentism and Probability

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Questions and Answers

What does a probability of 0 indicate?

The event is guaranteed to occur

The event is likely to occur

The event occurs half the time

The event never occurs (correct)

What is the primary idea behind the law of large numbers?

Probabilities become less stable with more trials

Independent events become related as trials increase

Outcomes are random with no predictable pattern

Averages become more stable as the number of trials increases (correct)

Which rule of probability states that the sum of all possible outcomes must equal 1?

Rule A

Rule C

Rule B (correct)

Rule D

What does personal probability express?

An individual's judgment about the likelihood of an outcome Signup and view all the answers

Which statement best describes the myth of short run regularity?

Randomness does not exhibit regular patterns in the short run Signup and view all the answers

What is a sampling distribution?

The values a statistic takes in repeated samples Signup and view all the answers

Which of the following represents a misunderstanding of the law of averages?

Gamblers believe they are 'due' for a win after a series of losses Signup and view all the answers

In the context of probability, what does a probability of 1 signify?

The event is guaranteed to occur Signup and view all the answers

What is a density curve used to represent?

The numerical distribution of a histogram Signup and view all the answers

How is the probability of an event calculated?

By adding the probabilities of the outcomes that make up the event Signup and view all the answers

Which statement accurately describes the relationship between odds and probability?

Odds are calculated as the probability of occurrence divided by the probability of non-occurrence Signup and view all the answers

When is simulation most appropriately used?

When convenience and time efficiency are desired Signup and view all the answers

What is one of the basic assumptions regarding independence in statistics?

The outcome of one phenomenon does not alter the outcomes of another Signup and view all the answers

What is the first step in conducting a simulation?

Give a probability model Signup and view all the answers

If the probability that Taylor Swift and Travis Kelce are not dating is 0.20, what is the probability that they are dating?

0.80 Signup and view all the answers

Which probability rule applies when determining the likelihood of TAMU or Mississippi winning the SEC West title?

Addition Rule Signup and view all the answers

What is the expected probability of a heads landing after a heads in a coin toss?

0.5 Signup and view all the answers

Which statement best describes the concept of correlation?

Correlation is limited to measuring linear relationships. Signup and view all the answers

What is the purpose of a tree diagram in probability?

To show possible outcomes and probabilities at each stage. Signup and view all the answers

How do you calculate expected values?

Determine the outcomes and multiply each by its probability and then sum the products. Signup and view all the answers

What does the law of large numbers suggest about observed outcomes?

The mean of the observed outcomes will approach the expected value. Signup and view all the answers

What is a key characteristic of the 'pari-mutuel' system?

Payouts depend on the total amount wagered and no fixed payouts exist. Signup and view all the answers

What does informed consent ensure in psychological research?

Participants are fully aware of the risks and can withdraw at any time. Signup and view all the answers

What does the Institutional Review Board (IRB) primarily do?

Review studies to protect participants from harm. Signup and view all the answers

Study Notes

Frequentism

Observing outcomes and patterns of events if they are repeated over and over again
Truths of probability:
- Chance behavior is unpredictable in the short run but has a regular and predictable pattern in the long run
- Individual outcomes are uncertain but there is a regular distribution of outcomes in a large number of repetitions
Probability is a number between 0 and 1
Probability describes the proportion of times the outcome would occur in a very long series of repetitions
- It represents the long-term regularity of random behavior
- 0 = outcome never occurs, 1 = outcome always occurs, 0.5 = outcome happens half the time in a very long series of trials

Myth of Short Run Regularity

The idea of probability is that randomness is regular in the long run, not in the short run
- Example: Coin toss - half heads/half tails in the long run

Myth of Law of Averages

Assumes that independent events are related and more likely to occur depending on each other (false)
- Example: Gamblers believe they are 'due' for a win after 5 losses

Law of Large Numbers

In a large number of independent repetitions of a random phenomenon, averages or proportions are likely to become more stable as the number of trials increases
- The sums/counts are likely to become more variable
The outcome of one trial does not change the probabilities for the outcomes of any other trials
- Trials have no memory

Personal Probability

A number between 0 and 1 that expresses an individual's judgment of how likely a particular outcome is

Probability Model

Describes all possible outcomes for a random phenomenon
A statistical model that describes all the possible outcomes and says how to assign probabilities to any collection of outcomes

Rules of Probability

Rule A: Any probability is a number between 0 and 1
Rule B: All possible outcomes must sum to 1
Rule C: The probability that an event does not occur is 1 minus the probability that the event does occur
Rule D: If two events have no outcomes in common, the probability that both events occur is the sum of their individual probabilities

Sampling Distribution

What values the statistic takes in repeated samples from the sample population
Assigns probabilities to the values the statistic can take

Density Curve

Graphical representation of a numerical distribution of a histogram (review from previous exam)
Used to describe sampling distributions when there are many possible values
Often described by a density curve such as a Normal curve

Probability of Events

Add the probabilities of the outcomes that make up the event

Odds of Events

The odds of "Y to Z" that an event occurs corresponds to a probability of Y/(Y+Z)
- You lose Y times and win Z times
Odds represent the probability that the event will occur divided by the probability that the event will not occur

Simulation

Allows us to imitate repeated trials of chance events
Uses random digits from a table or computer software to imitate chance behavior

When to Use Simulation

For convenience! It is easier than doing the math yourself and faster than running many repetitions in the real world
Gives good estimates of probabilities

Steps of Simulation

Give a probability model
Assign digits to represent outcomes
Simulate many repetitions

Assumptions of Independence

Knowing the outcomes of one phenomenon does not change the probabilities of the outcome of another
Each trial has no memory from one trial to another
Approaches:
- Apply the definition: For example, the probability of a coin landing heads after a heads should still be 0.5
- Check for correlation: Correlation should be = 0 (limited to linear relationships)
- Visualization: Scatterplots should show no patterns
- Know your study thoroughly (most commonly used by researchers)

Simple Random Phenomenon

Several independent trials with the same possible outcomes and probabilities for each trial

Tree Diagrams

A probability model in graphical form that shows stages and the possible outcomes and probabilities at each stage
Helpful by providing a visual representation of the probability model

More Complex Random Phenomenon

To simulate more complex random phenomena, string together simulations of each stage
May require varying numbers of trials or different probabilities at each stage, or stages that are not independent

Expected Values

An average of the possible outcomes
- Outcomes with higher probability count more

Calculating Expected Values

Determine all the possible outcomes
Determine the probability of each outcome
Multiply the outcomes and probabilities together
Add all the products

Expected Values - Law of Large Numbers

The mean of the observed outcomes approaches the expected value
The proportion of each possible outcome will be close to its probability
The average outcome will be close to the expected value
This demonstrates the long-run regularity of chance events

“Pari-mutuel” System

Payouts depend on the amount wagered
- No fixed amounts, so expected values cannot be calculated
Constants:
- The state keeps half the money bet
- Less risky for the state
Example: Casinos

IRB

Institutional Review Board
Reviews all studies in advance to protect participants from possible harm
Questions surrounding their workload and effectiveness in recent years

Participants must be informed about the nature of the study and any physical or psychological harm that it may bring
Participants give consent via their signature, and this can be retracted at any time during or after the study

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Description

This quiz covers the concepts of frequentism, emphasizing the long-term patterns observed in repeated events and the unpredictability of short-term outcomes. Key topics include the probability scale and common misconceptions such as the myth of short run regularity and the law of averages.