When I hear that a system has a one in a trillion (1,000,000,000,000) chance of failure, I immediately translate that in my mind to “So, optimistically the system has a one in a million (1,000,000) chance of failure.”
Extremely small probabilities are suspicious because they often come from one of two errors:
- Wrongful assumption of independence
- A lack of imagination
Wrongfully assuming independence
The Sally Clark case is an infamous example of a woman’s life being ruined by a wrongful assumption of independence. She had two children die of what we would call in the US sudden infant death syndrome (SIDS) and what was called in her native UK “cot death.”
The courts reasoned that the probability of two children dying of SIDS was the square of the probability of one child dying of SIDS. The result, about one chance in 73,000,000, was deemed to be impossibly small, and Sally Clark was sent to prison for murder. She was released from prison years later, and drank herself to death.
Children born to the same parents and living in the same environment hardly have independent risks of anything. If the probability of losing one child to SIDS is 1/8500, the conditional probability of losing a sibling may be small, but surely not as small as 1/8500.
The Sally Clark case only assumed two events were independent. By naively assuming several events are independent, you can start with larger individual probabilities and end up with much smaller final probabilities.
As a rule of thumb, if a probability uses a number name you’re not familiar with (such as septillion below) then there’s reason to be skeptical.
Lack of imagination
It is possible to legitimately calculate extremely small probabilities, but often this is by calculating the probability of the wrong thing, by defining the problem too narrowly. If a casino owner believes that the biggest risk to his business is dealing consecutive royal flushes, he’s not considering the risk of a tiger mauling a performer.
A classic example of a lack of imagination comes from cryptography. Amateurs who design encryption systems assume that an attacker must approach a system the same way they do. For example, suppose I create a simple substitution cipher by randomly permuting the letters of the English alphabet. There are over 400 septillion (4 × 1026) permutations of 26 letters, and so the chances of an attacker guessing my particular permutation are unfathomably small. And yet simple substitution ciphers are so easy to break that they’re included in popular books of puzzles. Cryptogram solvers do not randomly permute the alphabet until something works.
Professional cryptographers are not nearly so naive, but they have to constantly be on guard for more subtle forms of the same fallacy. If you create a cryptosystem by multiplying large prime numbers together, it may appear that an attacker would have to factor that product. That’s the idea behind RSA encryption. But in practice there are many cases where this is not necessary due to poor implementation. Here are three examples.
If the calculated probability of failure is infinitesimally small, the calculation may be correct but only address one possible failure mode, and not the most likely failure mode at that.