Probability and the lottery

The numbers drawn in the Israeli national lottery on 16th October were exactly the same as those drawn three weeks previously on 21st September. This has led to a storm of protest, with many suggesting that the lottery is somehow “fixed”.

In the Israeli lottery you pick 6 balls from a possible 37, and 1 from a possible 8. The chances of correctly picking 6 from 37 is 1 in 2324784.* Multiply this by the 1 in 8 chance of picking the second number gives a total probability of winning the jackpot in the Israeli lottery of 1 in 18598272.

You might think it illogical to play 1, 2, 3, 4, 5, 6 as numbers, on the grounds that a sequential sequence of numbers is less likely to occur than a random sequence such as 5, 11, 19, 23, 29, 43. But this is not true; the probability of any combination of numbers – sequential or otherwise – is the same. The lottery machine does not “know” what numbers have come up previously (or in previous weeks) and cannot set out to “avoid” creating sequential sequences.

(Selecting 6 balls from 49, as in the UK National Lottery)

The chance of the same numbers coming up two weeks in a row are exactly the same as any other combination of numbers.

Human beings are not good at generating random sequences, or at recognising truly random sequences; the tendency of human beings to see significance in randomness is called apophenia. This is what led to the protests in Israel: people saw significance (“cheating”) in the randomly selected lottery numbers.

Apophenia is more commonly associated with visual and audible stimuli, a specific type of apophenia called pareidolia. This tendency to see patterns in randomness is what leads to sightings of a face on Mars or religious figures in toast or wood grain.

In the image above the original “Face on Mars” image from 1976 is on the left, and a higher-resolution image of the same geological feature from 2001 is on the right.

* You can calculate this yourself using a scientific calculator with an “nCr” (combinations) button: 37C6 is 2324784. This is calculated by doing n!/r!(n-r)!.

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