16.6: Why Things Go Wrong

For a spacecraft to make it to the surface of Mars, many different subsystems must function properly. The computer, the radio, the rocket engines, and more must all work. If any of them fail, the entire mission could be ruined. In the case of the Space Shuttle Challenger, a problem with the O-rings was enough for catastrophic results.

Similar points apply to many other cases. For a computer or a car to work, all the parts must work. For the human body to remain healthy, all the various “systems” need to work; the cardiovascular system, the immune system, the nervous system, and many other things must each function well. In general, a complicated system will only function properly if its parts each function properly.

To say that all the subsystems of something must work is to say that subsystem one must work, and subsystem two must work, and subsystem three must work, and so on. Your computer’s central processor, and its hard drive, and its monitor, and . . . all must work for the computer to work. This means that a conjunction must be true. And as we saw in the previous section, the probability of a conjunction is usually lower than the probabilities of its conjuncts.

Imagine a spacecraft consisting of five subsystems. If any one of them fails, the entire mission will fail. Suppose that the probability that each subsystem will work is 9, and that the performance of each is independent of the performance of the others (this simplifying assumption won’t really be the case, but it doesn’t affect the present point in any relevant way). Then the probability that all five of the subsystems will work is .9 x .9 x .9 x .9 x .9 (= .9₅), which is a bit less than 6. If there were seven subsystems (all with a 9 probability of working), the probability that all seven would function properly is less than .5.

Even if the probability that each part of a complex system will function correctly is .99, if there are enough systems, failure somewhere along the line is likely; the more components there are, the more the odds against success mount up (this explains why spacecraft typically include backup systems).

We can make the same point in terms of disjunctions. A chain is no stronger than its weakest link; if any of them break, the entire chain gives way. If the first one breaks, or the second, or the third, . . . the chain is broken. Many things are like chains; they can break down in several different ways. In many cases, the failure of one part will lead to the failure of the whole.

In our imaginary spacecraft, the failure of subsystem one or of subsystem two or of subsystem three or . . . can undermine the entire system. This is a disjunction, and the probabilities of disjunctions are often larger than the probabilities of any of their disjuncts (this is so because we add probabilities in the case of disjunctions).

The lessons in this section also apply to things that occur repeatedly over time. Even if the probability of something’s malfunctioning on any occasion is low, the cumulative probability of failure over a long stretch of time can be moderate or even high.

Contraceptives are an example of this. On any given occasion, a contraceptive device may be very likely to work. But suppose that it fails (on average) one time out of every 250. If you use it long enough, there is a good chance that it will eventually let you down. To take another example, the chances of being killed in an automobile accident on any trip are low, but with countless trips over the years, the odds of a wreck mount up.

There is considerable evidence that people tend to overestimate the probabilities of conjunctions (thinking them more likely than they really are) whereas they underestimate the probabilities of disjunctions (thinking them less likely than they really are). As a result, we tend to overestimate the likelihood of various successes while we underestimate the likelihood of various failures.

As before, you don’t need to know precise probability values to appreciate these points. You may know that a contraceptive is likely to work, but that there is a non-negligible chance it will fail. This tells you that, over time, there is a very real chance of its failing.