BRM - Chapter 18 Probability Models

Probability model - definition

A description of a random phenomenon that lists all possible outcomes and specifies how to assign probabilities to any collection of outcomes (events).​

Event - definition

A collection (set) of outcomes from a random phenomenon; the probability of an event is found by adding the probabilities of the outcomes in it.​

Example of a probability model - marital status

For women aged 25-29: Never married 0.478, Married 0.476, Widowed 0.004, Divorced 0.042. These four outcomes and probabilities form a probability model.​

Probability rule A - range of probabilities

Any probability is a number between 0 and 1; 0 means never occurs, 1 means always occurs, 0.5 means occurs in half the trials in the long run.​

Probability rule B - total probability

All possible outcomes together must have total probability 1; the sum of the probabilities for all outcomes in a probability model is exactly 1.​

Probability rule C - complement rule

The probability that an event does not occur is 1 minus the probability that it does occur: P(not A) = 1 − P(A).​

Probability rule D - addition rule for disjoint events

If two events have no outcomes in common, the probability that one or the other occurs is the sum of their individual probabilities.​

Legitimate probability assignment - conditions

Any assignment of probabilities to individual outcomes that satisfies Rules A (between 0 and 1) and B (sum to 1) is legitimate.​

Complement example - not married

If P(married) = 0.476, then P(not married) = 1 − 0.476 = 0.524 using the complement rule.​

Event as a collection of outcomes - example

"Not married" is the event {never married, widowed, divorced}; its probability is the sum 0.478 + 0.004 + 0.042 = 0.524.​

Personal probabilities - idea

Subjective or personal probabilities express an individual's judgment about how likely an outcome is, such as experts' beliefs about Super Bowl winners.​

Coherent personal probabilities

Personal probabilities must still obey Rules A and B; if they do not, they are called incoherent because they do not make sense together.​

Incoherent personal probabilities - definition

A set of personal probabilities that violates the rules (not between 0 and 1 or not summing to 1 over all outcomes) is incoherent.​

Odds - definition

Betting odds of "Y to Z" mean a bet of $Z will pay $Y if the team wins; for a fair bet, these odds correspond to a probability of Z/(Y + Z).​

Converting odds to probability

Odds of Y to Z correspond to probability = Z / (Y + Z).​

Example - Patriots odds 6 to 1

Odds 6 to 1 that the Patriots win correspond to a probability of 1 / (6 + 1) = 1/7.​

Example - 49ers odds 14 to 1

Odds 14 to 1 that the 49ers win correspond to a probability of 1 / (14 + 1) = 1/15.​

Interpreting Super Bowl 53 probabilities

The listed "1/7, 1/11, 1/13, …" values are best interpreted as personal probabilities that can change as the season progresses.​

Sampling distribution - definition

The distribution of a statistic that tells what values the statistic takes in repeated samples from the same population and how often it takes those values.​

Sampling distribution as probability model

A sampling distribution assigns probabilities to the values a statistic can take in repeated random samples.​

Density curve as probability model

A density curve (like a Normal curve) can be used as a probability model by assigning probabilities as areas under the curve; total area (and probability) is 1.​

Normal curve and sampling distributions

A Normal curve often approximates the sampling distribution of a statistic (such as a sample proportion), assigning probabilities to its possible values.​

Long-run interpretation of probability

Probability describes the pattern of outcomes in the long run, over very many repetitions of the random phenomenon.​

Example - sample proportions and Normal curve

For many SRSs, the histogram of sample proportions can be approximated by a Normal curve, which then assigns probabilities to ranges of sample proportions.​

Empirical confirmation - SRS example

In one example, 94.3% of 1000 SRS sample proportions fell in an interval that matched closely with the probability calculated from the Normal curve.​

Probability model for sampling - key idea

Choosing a random sample and computing a statistic is itself a random phenomenon, and its pattern is described by a probability model (sampling distribution).​

Two types of probability models - outcomes

One type assigns a probability to each individual outcome (like the four marital statuses) and uses sums of probabilities for events.​

Two types of probability models - density curves

The second type uses a density curve (such as a Normal curve) where areas under the curve correspond to probabilities for ranges of values.​

Probability rules apply to all models

All legitimate probability assignments, whether data-based or personal, obey the same probability rules, so the mathematics of probability is always the same.​

Odds and probability summary

Odds of Y to Z that an event occurs correspond to a probability of Z/(Y + Z); this connects betting language to probability model