Section 7.1: Section 7.1: Chance and ProbabilityChance and Probability

ProbabilityProbabilityChance behavior is unpredictable

in the short run (you probably can’t guess what number will be drawn, which side the coin will fall on, etc.), but chance behavior has a regular and predictable pattern in the long run.

What does that mean?

Probability of tossing Probability of tossing heads…heads…

ProbabilityProbabilityWith a coin toss, “probability 0.5”

means “occurs half the time in a very large number of trials”.

The idea of probability is empirical… based on data rather than theorizing.

ProbabilityProbabilityA probability experiment is a trial

through which specific results (counts or measurements) are obtained.

The result of a single trial is an outcome.

The set of all possible outcomes is the sample space.

An event consists one or more outcomes and is a subset of the sample space. If it has only one outcome, it is referred to as a simple event.

ExampleExampleFor quality control, you randomly

select a computer chip from a batch that has been manufactured that day. Event A is selecting a defective chip.◦The experiment is selecting the chip to

inspect.◦The possible outcomes are defective

or not defective, which together make up the sample space.

◦Since there is only one possible outcome, it is a simple event.

Probability TheoryProbability Theory◦Mathematical study of randomness.◦Originated by Pierre de Fermat and

Blaise Pascal◦People wanted to see the “fair value”

of bets (how should I bet).

Law of AveragesLaw of AveragesIt’s a myth.People think if the same outcome

occurs over and over, a different outcome will occur next (heads 5 times in a row…it must be tails next.)

Because flipping a coin each time has no impact on the next flip (coins don’t have memories), these events are mutually exclusive (they have no impact on each other)

Law of AveragesLaw of Averages

An exampleAn exampleA few years ago, “Dear Abby” published in

her advice column a letter from a distraught mother of 8 girls. It seems that she and her husband had planned to limit their family to four children. When all four were girls, they tried again—and again, and again. After 7 straight girls, even her doctor had assured her that “the law of averages was in our favor 100 to 1”. Unfortunately for this couple, having children is like tossing coins. Eight girls in a row is highly unlikely, but once 7 girls have been born, it is not at all unlikely that the next child will be a girl—and it was.

Was it chance or Was it chance or causation?causation?In 1979, 2 of the 8 town wells serving

Woburn, MA, were found to be contaminated with organic chemicals. Alarmed citizens began counting cancer cases. Between 1964 and 1983, 20 cases of childhood leukemia were reported. This is an unusual number of cases for this rather rare disease. The residents believed the well water had caused the leukemia and proceeded to sue the 2 companies held liable for the contamination.

The results of an The results of an investigationinvestigationInvestigators obtained complete data

on everyone who had lived in the neighborhoods during the periods in question and to estimate their exposure to the suspect drinking water.

They looked for other factors such as occupational exposure to toxins and smoking.

The verdict: There was evidence of an association between drinking water from the 2 wells and developing childhood leukemia.

Personal ProbabilityPersonal ProbabilityPersonal Judgment (how likely

something is to happen based on your opinion).

We base our decisions on them.Because they are opinions, they

aren’t right or wrong.Still a number between 0 and 1.

Probability is used to Probability is used to describe riskdescribe riskWe feel safer when risk seems under

our control (risk while driving, as opposed to risk of asbestos or terrorism).

It’s hard to comprehend very small probabilities (we overestimate small risks and underestimate large risks).

Probabilities for risks like asbestos are only estimates (we should suspect that experts have underestimated the risk).

WorkWorkGroup Work:

◦White Book: Page 125, #1-10

Homework: ◦Green Book: Pages 411..422 #4, 8,

12-14, 18