● Uncertainties abound in life. (e.g. What's the gas price going to be next week? Is your lottery ticket going to win the jackpot? What's the presidential approval rate going to be in the next opinion poll?)

– Moreover, the world is fundamentally uncertain according to quantum physics

● Characterizing uncertainty with probabilities – Definitions

● Relative frequency interpretation

● Personal/subjective probabilities

– Probability rules – Probability distributions (sampling distributions being

important special cases for next time)

● Expectation of a probability distribution; Law of large numbers

Probability and Probability Distributions

Relative Frequency Probability

● Relative-Frequency Interpretation

– The proportion of time the event in question occurs over the long run.

– “Long-run relative frequency”

Relative Frequency Probability: e.g. Meaning of P(head)=1/2

Relative-Frequency Probabilities

● Can be applied when the situation can be repeated numerous times (conceptually) and the outcome can be observed each time.

● Relative frequency (proportion of occurrences) of an outcome settles down to one value over the long run. That one value is then defined to be the probability of that outcome.

● The probability cannot be used to determine whether or not the outcome will occur on a single occasion (it is a long-run phenomenon).

Relative Frequency Probability: doesn't predict outcome in a single experiment

Personal/Subjective Probability

● Personal-Probability Interpretation– The degree to which a given individual believes

the event in question will happen.

– Personal belief

– E.g. The probability that your favorite candidate will win the next presidential election

Personal or Relative Frequency Probabilities?

● The probability that a lottery ticket will be a winner.

● The probability that you will get an A in this course.

● The probability that the 7 a.m. flight from San Diego to New York will be on time on a randomly selected day.

Probability Rules

● Four simple, logical rules which govern the behavior of well defined probabilities

● Rule 1: A probability is a number between 0 and 1.● Rule 2: All possible outcomes together must have

probability 1.● Rule 3: The probability that an event does not

occur is 1 minus the probability that the event does occur. (e.g. If the probability that a flight will be on time is 0.7, then the probability it will be late is 0.3)

● Rule 4: The probability that one or the other of two mutually exclusive events occurs is the sum of their individual probabilities. – e.g. Age of woman at first child birth: under 20:

25%; 20-24: 33%; 25+: ? (1-.25-.33=42%)

Unmarried

Couples

Married

Couples

Avoid Being Inconsistent

Married with

Children

(Probability of married with children must not be greater than the probability that the couple is married.)

All Couples

Risk and Relative Risk● Often the term risk is used in place of

probability. And relative risk means the ratio of two probabilities.

● e.g. Based on experimental data, the risk of continuing to smoke is:– Nicotine patch: .53– Placebo patch: .80

– Relative risk of continuing to smoke when using the placebo patch compared with when using the nicotine patch is 1.5 ( = .800/.533). That is, the risk of continuing to smoke when using the placebo patch is 1.5 times the risk when using the nicotine patch.

Relative Risk: What is the Baseline?

● When the baseline is missing, the substantive meaning of the relative risk is unclear.

● E.g. Premature-birth risk found higher for teens: “The youngest girls [in the study] , those aged 13 to 17, were 90 percent more likely than the women in their early 20’s to deliver prematurely.”

– The relative risk was 1.9.– But what is the absolute risk for women in

their 20’s which is used as the baseline?

Probability Distributions (or “Models”)

● A probability distribution describes all the possible values a random variable can take, and the probabilities associated with each outcome (or collection of outcomes.)

– e.g. Coin toss: P(head)=P(tail)=.5. – e.g. Distribution of sample means. (normal)

● Compare with empirical/observed distributions we've looked at through histograms, etc.

– In the case of sampling distributions, we often only observe one realization from the underlying distribution.

Probability Distribution● E.g. Education:

– On observed data, we can do frequency table/histogram/bar chart.

– Ask the question: “What's the education level of a randomly selected individual?” and we have a random variable taking one of the 5 possible values with associated probabilities reflected in the table below.

Expected Value

● For observed data, we discussed characteristics of the distribution such as the mean and the standard deviation

● Can similarly characterize a probability distribution. Here we discuss the expectation, counterpart of the mean.

● Idea is that while we cannot predict individual outcomes, we can predict what happens (on average) in the long run.

● e.g. Tickets to a school fund-raiser event sell for $1. One ticket will be randomly chosen, the ticket owner receives $500. They expect to sell 1,000 tickets. Your ticket has a 1/1000 probability of winning.

– Two outcomes:

● You win $500, net gain is $499.● You do not win, net “gain” is $1.

– Your expected gain (expected value) is ($499)(0.001) + ($1)(0.999) = $0.50.

– long term, you lose an average of $0.50 each time (conceptually) you enter such a contest. (the school needs to make a profit!)

Expected Value

E.g. Make a decision, which would you choose?

● First alternative: EV=$240, no variation.● Second alternative:

EV=($1000)(0.25) + ($0)(0.75) = $250● If choosing for one trial:

– option (2) can either maximize the potential gain ($1000) or minimize it ($0)

– option (1) guarantees a gain ($240)– Decision influenced by risk attitude

● If choosing for many (say 500) trials:– option (2) will maximize expected gain

(will make more money in the long run)

(1) A gift of $240, guaranteed.(2) A 25% chance to win $1,000 and a 75% of getting

nothing.

Expected Value

E.g. Make a decision, which would you choose?

(1) A sure loss of $740.(2) A 75% chance to lose $1,000 and a 25% to lose nothing.

● First alternative: EV=-$740, no variation.● Second alternative:

EV=-[($1000)(0.75) + ($0)(0.25)] = -$750● If choosing for one trial:

–option (2) can either minimize the potential loss ($0) or maximize it (-$1000)–option (1) guarantees a loss–Decision influenced by risk attitude

● If choosing for many trials:–option (1) will minimize expected loss (will lose less money in the long run)

Expected Value

The Law of Large Numbers● Another way of interpreting the expected value

● Central idea: sample mean approaches population mean as N gets large (e.g. Sample mean income approximates population mean; proportion of heads in coin tosses approaches 0.5 in large number of tosses.)

● See “expected value” applet

● The higher the variability in the population, the larger the sample needed (more later: sample mean distributed normal with variance proportional to s)

● Simulation: for distributions with complex/unknown theoretical expectations, expected values can be approximated by simulating many random draws from the distribution and finding the average of all of the outcomes (e.g. quantities of interest from a model—beyond this course)

The Law of Large Numbers:The Case of Gambling

● The “house” in a gambling operation is not gambling at all:– the games are designed so that the gambler has

a negative expected gain per play– each play is independent of previous plays, so the

law of large numbers guarantees that the average winnings of a large number of customers will be close the the (negative) expected value

– thus the cheap buses/entertainment/accommodation to attract gamblers!