These notes capture the key concepts, discussions, and important information from the class sessions. They are intended to provide a comprehensive summary of the material covered, including lecture highlights, significant topics, and any additional insights provided by the instructor.
Chapter 17
The Idea of Probability
● Frequentism
○ What would happen if we did this many times?
● The truth about probability
○ In the short run, chance behavior is unpredictable
○ In the long run, pattern = regular and predictable
● Random
○ Individual outcomes are uncertain (by chance)
○ But, the distribution of outcomes is predictable in a large number of repetitions
● Probability
○ A number between 0 and 1
○ Describes the proportion of times the outcome would occur in an infinite series of repetitions
○ Language to describe the long-term regularity of random behavior
● Probabilities
○ 0 → never occurs (happens 0% of the time)
○ 1 → happens on every repetition (100% of times)
○ 0.5 → happens half (50%) of the times in an infinite series of trials
Myths about Chance Behavior
● The myth of short run regularity
○ Randomness is only regular in the long run
○ However, our brains are constantly looking for patterns, even in the short run…
■ When things don’t look random in the short run, we look for some explanation other than
chance variation
● The myth of the law of averages
○ Roulette wheel
■ I’ve won the last three games → I must be due for a win soon, right?
○ Flipping a coin
■ 6 heads in a row → a tails has to happen soon, right?
○ PBSI 245 in-class extra credit
■ It’s been awhile → surely next class will have an in class extra credit, right?
○ In each of these scenarios, the implicit assumption is that things must “average out”
○ We are expecting the probability of the next outcome to change based on the previous outcomes
○ There is no such thing as the “law of averages” but there is the “law of large numbers”
■ In a large number of independent repetitions of a random phenomenon (like coin tossing),
averages or proportions are likely to become more stable as the number of trials increases
● Note: in contrast, sums or counts are likely to become more variable
Personal Probabilities
● Layperson
○ Probability actually is just a judgment
● Decisions made based on these judgments
○ Ex: we take the bus to campus because we think the probability of finding a parking spot is low
● Personal probability of an outcome
○ A number between 0 and 1
○ Expresses an individual’s judgment of how likely a particular outcome is
Stats in Summary
● Some things in the world, both natural and of human design, are random
, ● That is, their outcomes have a clear pattern in very many repetitions even though the outcome of any one
trial is unpredictable
● Probability
○ The long term regularity of random phenomena
● Features
○ Based on very many repetitions on which that outcome occurs (frequentism)
○ A number between 0 (the outcome never occurs) and 1 (always occurs)
○ Empirical
■ We can test it with data
● Probability
○ Difficult for humans to understand
○ Describes only what happens in the long run
● Short runs of random phenomena
○ May appear to have patterns → easy to get tricked
● Our brains are biased towards patterns
○ A helpful trick for learning things like language
○ Unhelpful when it comes to perceiving truly random events
○ Also leads to us engaging in certain logical fallacies
● Personal probabilities
○ An individual’s personal judgment of how likely outcomes are
○ Can be numbers between 0 and 1
● Inconsistency across people
○ Different people, different personal probabilities
● Probabilities based on data
○ May conflict with personal probabilities
○ May not be collectible at all, or at least very hard to estimate
Chapter 18
Probability Models
● A probability model describes
○ All possible outcomes for a random phenomenon
■ Ex: sampling a person from a population at random to determine their marital status
○ How to assign probabilities to any collection of outcomes (an “event”)
Probability Rules
● Any probability is a number between 0 and 1
○ Any proportion/probability is a number between 0 and 1
○ An event with probability 0 never occurs
○ An event with probability 1 occurs every time
○ An event with probability 0.5 occurs in half the trials in the long run
● The sum of probabilities of all possible outcomes must have a probability of 1
○ Some outcomes must occur on every trial
● The probability that an event does not occur is 1 minus the probability the event does occur
○ If an event occurs in 70% of all trials, it fails to occur in the other 30%
■ This possibility is called the complement
○ The probability that an event does occur and the probability that it does not occur is 100%, or
1 → P (event A) + P (not event A) = 1
● If two events have no outcomes in common, the probability that one or the other occurs is the sum of their
individual probabilities
○ Such events are called mutually exclusive
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