The UK National Lottery began on 19th November 1994 and there had been 1240 draws up to 20th October 2007. The jackpot prize is won by choosing in advance the 6 numbers that will be drawn from a set of balls numbered from 1 to 49. We can use the history of the lottery to illustrate many aspects of the theory of probability: how each draw is individually unpredictable, and yet the overall history shows predictable patterns; how a `league table' of numbers can be created that appears to show some numbers are preferentially drawn, and yet the table is completely spurious; how to test whether the balls are truly being drawn at random; how extremely unlikely events will occur if you wait long enough, and so on.
Starting from 1994, note how the 'leader' changes, until one number seems to gain a substantial lead.
If you click on 'Show histogram', you can create the current distribution showing how often each of the 49 numbers has come up. Press 'Start dropping' to see how that histogram arises. The distribution seems quite spread out, with some numbers appearing much more often than others, but in fact this apparent spread should be purely due to chance. Lottery Expectations considers what sort of distribution of total appearances of each number we would expect when lottery balls are chosen at random.
What about the gaps between numbers?
Using the animation above, work backwards from October 2007 and see how long you have to wait until the last number appears. Do you think this is surprising? We can use probability theory and simulations to explore how long we have to wait for a number to come up.
The animation below shows the gap between each time a number comes up.
Can you see the longest gap that has occurred? Look carefully from the start of 2000. Do you think this is surprising? Lottery Expectations considers what sort of gaps between numbers we would expect when lottery balls are chosen at random.
Further reading and links
This is based on an idea by Fenton on considering the lottery results as a league table.
A spreadsheet with the full lottery history can be downloaded from the main UK National lottery site