Probability and risk increasingly permeate our lives. Like it or not, we must be able to assess the threats and opportunities that face us. Here's a random sampling of half a dozen hypothetical questions (with answers at the end) inspired by a variety of recent news stories.
1. It's impossible to say with any precision what risk the Washington area snipers posed to individuals in suburban Maryland and Virginia, but certainly the likelihood of being attacked was quite small — 13 victims out of about four million people in the affected area over three weeks.
Our psychology, however, leads us to be more afraid of what's unfamiliar, out of our control, dramatic, omnipresent, or is the consequence of malevolence. On all these counts, the snipers were more terrifying than more common risks.
Still, let's consider one of these more common risks. How many traffic fatalities can be expected to occur in any given three-week period in the United States? How many in an area the size of suburban Washington?
2. Early in the sniper case the police arrested a man who owned a white van, a number of rifles, and a manual for snipers. It was thought at the time that there was one sniper and that he owned all these items, so for the purpose of this question let's assume that this turned out to be true.
Given this and other reasonable assumptions, which is higher — a.) the probability that an innocent man would own all these items or b.) the probability that a man who owned all these items would be innocent?
3. The Anaheim Angels and San Francisco Giants were in this year's World Series. The series ends, of course, when one team wins four games.
Is such a series, if played between equally capable opponents, more likely to end in six or seven games?
4. The rules of the series stipulate that team A plays in its home stadium for games 1 and 2 and however many of games 6 and 7 are necessary, whereas team B plays in its home stadium for games 3, 4, and, if necessary, game 5.
If the teams are evenly matched, which team is likely to play in its home stadium more frequently?
5. Eleven million people went to the polls recently in Iraq and, the Iraqi news media assure us, 100 percent of them voted for Saddam Hussein for president. Let's just for a moment take this vote seriously and assume that Hussein was so wildly popular that 99 percent of his countrymen were sure to vote for him and that only 1 percent of the voters were undecided. Let's also assume that these latter people were equally likely to vote for or against him.
Given these assumptions, what was the probability of a unanimous 100 percent vote?
6. Politics in a democracy is vastly more complicated than it is under dictatorships. Witness the upcoming elections here.
What is the probability that the Republicans, the Democrats, or neither will take control of the Senate on Nov. 5?
Answer to 1. There are approximately 40,000 auto fatalities annually in this country, so in any given three-week period, there would be about 2,300 fatalities. The area around Washington has a population of about four million, or 4/280 of the population of the U.S., so as a first approximation, we could reasonably guess that 4/280 times 2,300, or about 30 auto fatalities, would occur there during any three-week period. Attention must then be paid to the ways in which this area and its accident rate are atypical.
Answer to 2. The second probability would be vastly higher. To see this, let me make up some illustrative numbers. There are about four million innocent people in the area and, we'll assume, one guilty one. Let's estimate that 10 people (including the guilty one) own all the three of the items mentioned above. The first probability — that an innocent man owns all these items — would be 9/4,000,000 or less than 1 in 400,000. The second probability — that a man owning all three of these items is innocent — would be 9/10. Whatever the actual numbers, these probabilities usually differ substantially. Confusing them is dangerous (to defendants).
Answer to 3. For the World Series to last 6 or 7 games, it must last at least 5 games, at which point one team would be ahead 3 games to 2. If the team that is ahead wins the 6th game, the Series is over in 6 games. If the team that is behind wins the 6th game, the Series goes to 7 games. Since the teams are equally matched, the Series is equally likely to end in 6 or 7 games.
Answer to 4. The solution requires that we use a bit of probability theory. Doing so, we find that, on average, team A will play 2.9375 games at its home stadium and team B 2.875 games at its home stadium. Thus team A is a bit more likely to play at home.
Answer to 5. Even given the absurdly generous assumptions above, there would be 110,000 undecided voters (1 percent of 11 million). The probability of a 100 percent vote is thus equal to the probability of flipping a fair coin 110,000 times and having heads come up each and every time! The probability of this is 2 to the power of minus 110,000, or a 1 preceded by more than 30,000 0's and a decimal point. This would be the cosmic mother of all coincidences!
Answer to 6. As of this writing the Democrats hold a one vote edge in the Senate, and there are a number of races too close to call. Significant consequences will surely flow from small, but unpredictable factors so my prediction won't be ready until Wednesday, Nov. 6.
Professor of mathematics at Temple University and adjunct professor of journalism at Columbia University, John Allen Paulos is the author of several best-selling books, including Innumeracy, and the forthcoming A Mathematician Plays the Market, which will be published in the spring. His Who’s Counting? column on ABCNEWS.com appears the first weekend of every month.