animation

None of us are going to last for ever. Our prospects depend on our sex, our age, our lifestyle, our genes, and many other personal factors both known and unknown. Even with all this information we're all uncertain about the exact date of our death, but by looking at large groups of people who are like us, we can count how many die each year and so get an idea of the risks we face and how long we might live. Our risks can be summarised in different ways which are shown in the animation below.

In this animation, the computer simulates a number of people $N$ choosing a number at random between 1 and $(\frac{N}{2})^2$. So, for example, it will simulate 400 people choosing a number between 1 and 40000.
How often would you expect it to pick the same number twice?

The Premier League is the main English football league, with 20 teams each playing a home and away against each other team making, 38 matches for each team in a season, and 380 matches altogether. Teams are awarded 3 points for a win, 1 for a draw, 0 for losing, and the league position is decided on total points, with equal points decided by goal difference (goals for minus goals against). At the end of the season the bottom 3 teams are relegated.

National lottery shows how many times each of the numbers has come up in the main National Lottery draw, and what were the gaps between appearances of each number. Here we look at whether the observed distribution of the number of times each of the 49 numbers has come up fits with what would be expected with a truly random draw, and whether the gaps also correspond to what might be expected.

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.