Introduction to Frequentism and Probability

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

1. Give a probability model
2. Assign digits to represent outcomes
3. 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

1. Determine all the possible outcomes
2. Determine the probability of each outcome
3. Multiply the outcomes and probabilities together
4. 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

Informed Consent

• 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

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.