Probability Intuition Game

About Gut Odds

Gut Odds poses a chance question: two dice are rolled, how likely is a sum of 7? A fair coin is flipped five times, how likely are at least three heads? You slide to the chance in percent, and the game scores you against the exact probability, rounded to a whole percent. After each round a short note explains how the right answer comes about.

Seven families of questions appear. The first levels contain dice sums, coin flips and cards. Later come the chance of at least one six in several rolls, drawing two marbles from a bag without replacement and, from level 7, the birthday question: how likely is it that two people in a group share a birthday?

How to play

1. Read the question and decide whether the chance is small, middle or large.
2. Slide to your estimate in percent. 0 percent is impossible and 100 percent is certain.
3. Lock in within the time limit, 16 seconds on level 1 and 12 seconds on level 10.
4. The note after your answer explains the right answer, for example how many of the 36 dice outcomes qualify.

A chance question is shown, like rolling dice or drawing cards. Slide to how likely you think it is, in percent. The answer is the exact probability, rounded.

How scoring works

The score depends on the distance in percentage points between your answer and the exact probability. A perfect answer gives 100 and the score falls to 0 at a distance of 38 points on level 1 and 18 points on level 10. So on level 10, an estimate that is 9 points off earns about 50.

The probabilities are computed exactly in the code, for instance 6 out of 36 for a sum of 7, 16 out of 52 for a heart or a king, and 1 − (5/6)^n for at least one six in n rolls. Fair dice and coins, a well shuffled deck and a random draw are assumed.

What it trains

The game trains probabilistic intuition, one of the areas where people are known to make systematic errors. Having a better feeling for chances helps with everyday risks, with reading news about statistics, and with games of chance.

It also shows how often our gut feeling is quite good for simple events and quite poor for compound ones, such as 'at least one' of several tries or a match among many pairs.

Tips for a higher score

• For dice sums, count the combinations. There are 36 outcomes, and 7 can be made in six ways, so it is the most likely sum.
• For 'at least one', compute the chance of none and subtract it from 100 percent. The chance of no six in three rolls is about 58 percent, so at least one six has about 42 percent.
• For drawing without replacement, multiply fractions: for 6 red and 3 blue marbles the chance of two reds is 6/9 times 5/8.
• For card questions, count the qualifying cards out of 52, and watch out for overlap, such as the king of hearts.
• Do not trust a feeling of 'it is due'. Fair dice and coins have no memory.

The idea behind it

Psychologists Daniel Kahneman and Amos Tversky showed in the 1970s and 1980s that people judge probabilities with shortcuts that are often useful and sometimes lead to systematic errors. Examples are the gambler's fallacy, the belief that a result is due after a streak, and the conjunction fallacy, which they described in 1983 with the Linda problem.

The birthday problem, which is often credited to Richard von Mises in 1939, has a surprising answer: in a group of only 23 people the chance that two share a birthday is about 50.7 percent, and with 50 people it is about 97 percent. The reason is that there are 253 pairs among 23 people, far more than most people expect.

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