Humans have been dealing with uncertainty forever, but the mathematics of probability is quite recent, and our intuition for it shows signs of that. People handle simple, familiar odds well, such as "likely rain", and make systematic mistakes in situations that are common in modern life: medical tests, risks in the news and games of chance. Knowing the traps is the quickest way to improve.
Trap 1: neglecting the base rate
A classic problem was posed in 1978 to staff and students of Harvard Medical School by Casscells, Schoenberger and Graboys. A disease affects 1 in 1,000 people. A test has a false positive rate of 5 percent and detects every true case. A person tests positive. What is the chance that the person has the disease? About half of the respondents answered 95 percent. The correct answer is about 2 percent: in 1,000 people, one has the disease and about 50 healthy people get a false positive, so only 1 of roughly 51 positives is a true case. People focus on the test result and forget how rare the disease is. This is base-rate neglect.
Trap 2: the gambler's fallacy
After five reds in a row on a roulette wheel, black feels "due". It is not, since the wheel has no memory. The most famous example happened at the Monte Carlo casino in August 1913, where the ball fell on black 26 times in a row and gamblers lost millions betting against the streak. In independent events the past does not change the odds. The opposite mistake is the "hot hand": believing that a streak will continue. For real skills such as basketball the evidence is more complex, but for coins, dice and lotteries the streak means nothing.
Trap 3: bad feelings for small numbers
People tend to treat a 1 percent chance as nearly zero, or to blow it up to a serious worry when the possible outcome is dramatic. Studies by Kahneman and Tversky on prospect theory found that small probabilities are overweighted, while large ones are underweighted. This is part of the reason why people buy lottery tickets and insurance for unlikely events, and why the risk of a plane crash feels larger than the risk of the drive to the airport.
Trap 4: coincidences are more common than they feel
The birthday paradox is the standard example. In a group of 23 people the chance that at least two share a birthday is just over 50 percent, and with 70 people it is above 99 percent. It feels wrong since we compare ourselves with others, 22 chances, and not all pairs, which number 253. The lesson is that with a large number of opportunities, even rare events happen to someone, often. A remarkable coincidence is almost always less remarkable than it feels.
Trap 5: randomness does not look random
If you ask people to write a random series of heads and tails, they alternate too often and avoid long runs. In a real series of 20 coin flips, a run of five or more of the same side is more likely than not. This is why a real random pattern looks "clumpy", and a pattern with a very even spread looks designed. Your gut feeling for what looks random is often not the same as what is likely.
What helps: natural frequencies
Gerd Gigerenzer and Ulrich Hoffrage showed in 1995 that most mistakes shrink when problems are presented in natural frequencies instead of percentages. "Out of 1,000 people, 1 has the disease, and 50 healthy people also test positive" is much easier than "the prevalence is 0.1 percent and the false positive rate 5 percent". The first can be answered at a glance: one of fifty-one. Whenever you have to judge a risk, translate it into a group of 100 or 1,000 people and count.
- Ask for the base rate: how common is this in general?
- Ask for the reference class: out of how many people or events?
- Think in absolute numbers: "from 2 in 1,000 to 3 in 1,000" instead of "50 percent more".
- Ask whether the events are independent. If so, the past does not matter.
Practising gut odds
Our Gut Odds game poses small chance questions with an exact answer, such as the chance of a sum of 7 with two dice or of at least three heads in five coin flips. You slide to a percentage, and a short note after each round shows how the right answer is counted. Over a few dozen rounds you find out where your feeling goes wrong, and counting the outcomes, as in the note, is the habit that replaces it.