# 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)!.

# Probability and the Grand National

Entropy is a very important concept in physics. When I was taught about entropy it was always related to probabilities, and the problem sheets we did often featured betting, horse races, greyhounds, etc. All this came back to me this weekend with the running of the Grand National.

It turns out that bookmakers are using entropy when they decide their odds, albeit unknowingly.

Imagine a greyhound race in which there are six dogs running. The bookmaker assigns each dog an equal probability of winning – one-sixth – and thus each dog has odds of six to one (6-1).

The problem for the bookmaker is that I can come along and bet an equal sum of money on each dog and be guaranteed to recoup my total bet, the bookmaker cannot make any money.

With longer odds the problem is even worse:

Now the bookmaker is guaranteed to lose money. A £1 bet on each dog for a total of £6 yields a win of £7 no matter what the result.

The obvious answer is to shorten the odds:

Now the bookmaker is guaranteed to make money, provided that all bets are evenly distributed amongst the dogs. If I bet £1 on each dog again, I can only win £5 and the bookmaker is guaranteed a £1 profit.

But this isn’t realistic: in real life there is usually a ‘favourite’, a dog generally considered more likely to win than the others. Therefore the bookmaker has to offer shorter odds on this dog, in order to avoid losing too much money and the opposite is true for the dog least least likely to win – the bookmaker can offer longer odds because he’s less likely to have to pay out.

But how can a bookmaker be sure that his odds have been calculated correctly? How can he be sure that, no matter what the outcome, he doesn’t end up too much out of pocket?

Which set of odds should he offer? The green or the purple? The green will result in paying out less money, but the purple will entice more customers to place a bet.

Unconciously, the bookmaker is calculating the entropy of the system. The more disordered the system, the greater its entropy. The greater its entropy, the greater the reward for the bookmaker. With six dogs each at odds of 6-1 the entropy is exactly 1. With the dogs at 5-1 the entropy is greater than 1; and at 7-1 the entropy is less than 1.

If we make a number of assumptions about how people bet, we can analyse the odds and calculate whether or not the bookmaker will make a profit.

1. A dog’s odds are related to its chance of winning.
2. Bets will be distributed amongst dogs according to their odds (e.g. people are more likely to bet on the favourite).
3. With longer odds, more people will bet.

The green odds yield an entropy of 1.157 and the purple odds yield an entropy of 0.900. Assuming each person bets £1 and that 49 people will bet on the green odds and 55 on the purple odds the bookmaker can expect to make \$6.66 on the green odds and lose £6.08 on the purple odds.

You can check yourself using the downloadable odds calculator (23.5kB, Excel). Change the bold values to check the outcome.

I worked it out and the Grand National’s entropy was 1.398, so Bookmakers should be happy!