Probability models

Choose a woman aged 25 to 29 years old at random and record her marital status. “At random” means that we give every such woman the same chance to be the one we choose. That is, we choose a random sample of size 1. The probability of any marital status is just the proportion of all women aged 25 to 29 who have that status—if we chose many women, this is the proportion we would get. Here is the set of probabilities:

Marital status: Never married Married Widowed Divorced
Probability: 0.478 0.476 0.004 0.042

This table gives a probability model for drawing a young woman at random and finding out her marital status. It tells us what are the possible outcomes (there are only four) and it assigns probabilities to these outcomes. The probabilities here are the proportions of all women in each marital class. That makes it clear that the probability that a woman is not married is just the sum of the probabilities of the three classes of unmarried women:

428

Table : TABLE 18.1 Probabilities of winning Super Bowl 50
Team Probability Team Probability
Green Bay Packers 1/5 Buffalo Bills 1/41
New England Patriots 1/7 Kansas City Chiefs 1/41
Seattle Seahawks 1/11 St. Louis Rams 1/51
Indianapolis Colts 1/13 New York Jets 1/61
Denver Broncos 1/16 New York Giants 1/86
Arizona Cardinals 1/21 Detroit Lions 1/101
Cincinnati Bengals 1/21 Houston Texans 1/101
Pittsburgh Steelers 1/21 New Orleans Saints 1/101
Atlanta Falcons 1/26 Oakland Raiders 1/101
Dallas Cowboys 1/26 San Francisco 49ers 1/101
Philadelphia Eagles 1/26 Tampa Bay Buccaneers 1/101
Carolina Panthers 1/31 Washington Redskins 1/101
Miami Dolphins 1/31 Cleveland Browns 1/201
San Diego Chargers 1/31 Tennessee Titans 1/201
Minnesota Vikings 1/36 Chicago Bears 1/251
Baltimore Ravens 1/41 Jacksonville Jaguars 1/301
Source: http://www.betvega.com/super-bowl-odds/ (as of September 21, 2015).

P(not married) = P(never married) + P(widowed) + P(divorced)

= 0.478 + 0.004 + 0.042 = 0.524

image Politically correct In 1950, the Russian mathematician B. V. Gnedenko (1912–1995) wrote a book, The Theory of Probability, that was popular around the world. The introduction contains a mystifying paragraph that begins, “We note that the entire development of probability theory shows evidence of how its concepts and ideas were crystallized in a severe struggle between materialistic and idealistic conceptions.” It turns out that “materialistic” is jargon for “Marxist-Leninist.” It was good for the health of Russian scientists in the Stalin era to add such statements to their books.

As a shorthand, we often write P(not married) for “the probability that the woman we choose is not married.” You see that our model does more than assign a probability to each individual outcome—we can find the probability of any collection of outcomes by adding up individual outcome probabilities.

Probability model

A probability model for a random phenomenon describes all the possible outcomes and says how to assign probabilities to any collection of outcomes. We sometimes call a collection of outcomes an event.