Thinking Involving Very Large and Very Small Quantities
For most of human existence, we lived in small groups and were unaware of things that happened outside of our own villages and a few nearby ones. Our brains evolved to be fairly effective in handling the kinds of decisions that were needed in that kind of setting. Things are very different today, when we routinely find out about things that happen everywhere in the world, and we make voting decisions that influence millions of people and billions of dollars. It seems clear that we can't make very good intuitive judgements when huge numbers are involved. While people would rarely say "ten" when they meant "one hundred", saying a million instead of a billion is a pretty common mistake. Intuitively a million is a lot more like a billion than ten is like one hundred, because our intuition has some grasp of ten and one hundred, but we have little grasp of what millions and billions involve.
Fortunately, there is often a way to make intelligent decisions involving big quantities. Use arithmetic! Typically we don't need more than multiplication and division to put things into terms we can deal with. If we are unwilling or unable to do the calculations, we should at least recognize that our intuitive judgement might be way off the mark.
Often we encounter large numbers when we hear amounts of money discussed at the national level. The Gross Domestic Product (GDP) of the United States is in the order of ten trillion dollars. That amount is meaningless to most people unless they can relate it to other things they know. Recently a friend was wondering about the 87 billion dollar expenditure President Bush had requested for the war in Iraq. Would this be a serious problem for the country? It sounds like a lot, but the United States is also a very big country. It is beyond ordinary human intuition to make direct sense of such a large quantity. Usually it is helpful to compare large numbers with each other. In this case it is useful to compare the GDP with the 87 billion dollar expenditure. One percent of ten trillion is 100 billion, so the Iraq expenditure is a little less than one percent of the GDP. Knowing this gives us a much better handle on how likely this is to affect the overall economy.
Sometimes we are given completely useless methods of looking at large amounts. If one dollar bills were laid end to end, a billion dollars would go around the earth three times. This isn't very useful. This is just expressing one incomprehensible quantity in terms of some even less comprehensible quantity. We should never be impressed by such gimmicks.
In general, everything done at a national or world level is too large for easy comprehension. If there were five thousand highway fatalities this year, is that a lot or a little? Suppose there were 30,000 new jobs created last month. How good is that? Normally we have to compare these numbers to what we would expect from past experience before we can decide whether such numbers are something to celebrate or are a disappointment.
At one time I was teaching in a college and also doing some programming for the computer center. The director of the computer center told me he had just heard a talk where the speaker pointed out that to play a perfect game of chess the number of possible moves that would have to be analyzed would be a one with 120 zeros. Aware of the amazing speed of computers (this was a while ago, so it was about .5 megahertz) he suggested I write a program to analyze all the possibilities to show how fast the computer was. I had to tell him that it would be a poor demonstration because the sun would have burned out and the earth would be long gone before the computation was done.
Let's imagine that the computer could analyze a thousand moves in a second (this would be optimistic). Then the length of time to do the calculation in seconds would simply be the total number of moves divided by the number of moves that could be analyzed in a second. That's one with 120 zeros divided by one with three zeros. The result is ten with 117 zeros. That's a lot of seconds. It turns out the number of years would be three with 109 zeros. The age of the universe so far is about 15 with nine zeros.
This kind of mistake is easy to make because both the number of moves and the speed of the computer are beyond our everyday comprehension. A little calculation clears it up quickly, however.
Very small amounts are just as hard to visualize as very large ones. One area these come up a lot are in probabilities. The odds of winning the lottery, hitting the jackpot on a slot machine, becoming a professional football player, getting hit by lightning, or being killed in a meteor strike are all very small, but we might want to have a realistic view of how likely these things are because their consequences are all pretty important.
Typically we overestimate small probabilities. One of the most common examples of this is people's concern about airplane crashes. From time to time we see news stories about airliner crashes, and there is usually a high fatality rate. The news stories will show smoldering wreckage and interviews with grief stricken family members and perhaps lucky survivors, so there is a strong emotional impact when we learn about crashes. As a result, many people are terrified of flying. Well, what are the odds that a commercial airliner will crash? In the year 2000, there was approximately one crash fatality for every 8,400,000 times a passenger boarded a scheduled commercial airliner. This turns out to be about as dangerous as driving 7.8 miles on a highway. In 2001, the odds were much worse because of the terrorist hijackings - a flight was about as dangerous as driving 50 miles.
Mentally we greatly overestimate the probability that a plane will crash, most likely because mentally we compare safe landings that we know about with crashes that we know about. The trouble is that we know about virtually all the crashes, since they are big news stories, but we know about only a tiny percentage of safe landings - perhaps when friends fly or we see some news involving a celebrity arriving at an airport. So if we want to make an intelligent decision about the safety of flying as opposed to the safety of going the same distance by car, we need to have some idea of the actual statistics rather than our intuition. Even with potential terrorists, travelling by commercial airliner is still far safer than going a the same distance by automobile.
Often when confronted with this information people will say "I know that's what the statistics say, but I can't help being terrified when on a plane, and the trip is so miserable I'd rather take my chances driving." This probably true for some people; however their fear might be decreased if they could improve their awareness of the number of safe landings. I have heard of a device that beeps every time an airplane makes a safe landing somewhere in the United States. People who are afraid of flying are encouraged to carry it around with them for a while so they can develop a sense for how common safe landings are. Since it beeps every second or so all day long, it usually doesn't take very long for the person to figure out that safe landings are incredibly common.
In the 1980's there was a lot of concern about terrorism in Europe, particularly following the hijacking of the cruise ship Achille Lauro in 1985. Tourism by Americans dropped off dramatically because of their fear. The reality was that terrorists actually killed very few tourists during this time period. Only one was killed on the Achille Lauro, although the other 700 people who were hostages on the ship no doubt had a harrowing experience. Most likely the chance of dying from food poisoning or a heart attack while on a vacation in Europe was far larger than being killed by a terrorist. However terrorist attacks always get extensive news coverage, while other sources of tourist deaths do not, so again people's intuition is likely to be fooled. Unfortunately, the illusion of danger helps terrorists, since the economic damage done by irrational fear is likely to far outweigh the physical damage the terrorists are capable of. If more people were able to correctly assess the dangers, the reduction in travel would be less, and not only would they get to enjoy their vacations, but the goals of the terrorists would be thwarted.
Large and small numbers combined
Often numbers too large for us to appreciate and numbers too tiny for us to appreciate interact as part of the same problem. This comes up in areas as diverse as playing the lottery, voting, and defending against meteor strikes.
In multi-state lottery games, there is often a huge top prize to the person or people who pick all the right numbers. In the Powerball lottery, the prize has sometimes exceeded one hundred million dollars. This is too huge for most of us to grasp intuitively. By the same token, the probability of winning is extremely tiny. This is far too small for us to grasp. The benefit of playing the game is tied up in both numbers. The more we can win, the more desirable it is to play. The smaller the chance of winning, the less desirable it is to play. The game is probably so popular because people mentally overestimate their chances of winning by a huge factor. As a result, the chance of winning such a large amount seems like it is worth the small ticket price.
One way to calculate the value of a ticket is to divide the amount won by the odds against winning. If we have one chance in a million of winning one hundred million dollars, then, on the average, spending a million dollars would get us a hundred million dollars back. In reality, the lottery would quickly go broke if this was the case. It turns out that most lotteries keep half the money paid for tickets and contribute the remainder to the prize pool, so in fact the average amount we win from each one dollar ticket is half a dollar. That, of course, includes all the smaller prizes in addition to the main jackpot.
Another case where there is a very tiny probability of a very large effect is the possible damage from a meteor striking the earth. The chances of a truly devastating strike like the one that killed the dinosaurs is very tiny. There hasn't been one that bad in 65 million years. On the other hand, if we do have a strike that large, everybody on earth might die, so the consequences are as bad as they could get. If we had strikes that killed six billion people approximately every 60 million years, that would work out to an average of one hundred people a year, probably not one of our biggest priorities, but worth a modest effort to prevent.
An internet article at http://impact.arc.nasa.gov/news/1998/dec/23.html states:
In the next 50 years there is a 1 in 2,000 chance of a 1 kilometre diameter asteroid impacting the Earth at a speed of around 70,000km/h. At this speed the object has more energy than its equivalent mass in TNT. The consequences would be global and catastrophic - this would not be an "extinction event" but perhaps one quarter of the world's population would die and civilization would collapse.
Using this estimate, we would have one chance in one hundred thousand in any year of losing one and a half billion people. That corresponds to 15,000 lives per year - a much more serious problem.
People might argue that a small chance of a huge disaster isn't the same as a large chance of a small disaster just because the average is the same. Perhaps, but I see no better way to compare the two situations. If twice as many people die, the problem seems twice as great. If it is only half as likely to happen, the problem is half as great. When dealing with chance events, the most sensible approach should be to try to reduce the average number of deaths per year.
A very different kind of case which involves small probabilities of large effects is that of voting. When we vote, it often seems discouraging because the election is very rarely decided by our particular vote. In an election that is reasonably evenly matched, say at the state level, we might have one chance in a thousand of casting the deciding vote. (If an election is not evenly matched, it is fair to say our vote has no chance of deciding the outcome.) Since the winner of the election is likely to influence how hundreds of millions of tax dollars are spent as well as changes in other laws that affect our lives, the overall impact is huge. It may not seem like a huge effect to me, if it only turns out to make, say, a 100 dollar difference. However in a state of five million people, that would correspond to an overall 500 million dollar difference. From a personal standpoint, I might have one chance in a thousand of making a 100 dollar difference, or an average of ten cents, but from the standpoint of the public as a whole, one chance in a thousand that our vote will make a 100 million dollar difference is very important. Thus, informed voting makes sense as a civic responsibility.
If we want our beliefs to be valid, it is important for us to realize that intuitive impressions are inadequate for making judgements about quantities that are so large or small that they are outside of our everyday experience. Sometimes we can bring things into perspective using a little research (to get some actual numbers) plus some simple arithmetic. If this isn't practical we should just recognize that we don't know the answer and avoid taking a strong stand on one side or the other of issues that depend on these quantities.