Oh dear, he’s going to try to teach us again…
Using Probability to Cheat the Lottery
The number that has come up the fewest number of times in the National Lottery in the UK is the number 20 (175 times). The number that has come up the most frequently is the number 44 (244 times). This is despite the fact we have every reason to believe each ball is equally likely to occur.
From this, can you guess which numbers you should pick next time you play? Should you pick the most frequently occurring numbers: 44, 38, 40, 23, 33, 39 on the grounds that they come up the most frequently? Do you really believe these balls are inherently more likely to be pulled out of the machine, in a repeatable way?
Or should you pick the numbers that have occurred the fewest number of times: 20, 13, 41, 16, 21, 15? If all the balls are equally likely to come out we expect the statistics to tend towards equal numbers, so are these numbers due?
What about the numbers that have gone the longest without being picked: 46, 12, 1, 27, 45, 35. Surely, every week they don’t get picked increases the likelihood of it being picked next time.
Have you decided which of these three options you’d side with?
If you selected option 1, no points. If you selected option 3, no points. But if you selected option 2, still no points. The fact is that historical trends have no impact on future consequences in a system like this. Each time all the balls are put back in and the probability doesn’t change. Some readers may be familiar non-experts with “Probability Theory”. Probability Theory is a method of determining how likely things are based on their historical trend; common things occur commonly, to over-simplify it.
Probability Theory Vs. The Principles of Probability
If historical patterns tell us nothing, what is Probability Theory about? See, historical patterns do tell us something. Historical patterns tell us the likelihood of a things happening, within certain confidence intervals.* But historical patterns don’t affect the probability, they are simply a method of understanding the chances. In fact, Probability Theory tells us that the chance of each ball coming out of the lottery machine is about equal, but not exactly equal. The principles of probability you learned in school tell us that Probability Theory came out as “about” equal because of the imperfections of statistics in action (i.e. the practical results).
*If Probability Theory tells us the probability of a thing happening 36/100, the truth could be anywhere between 30-42/100. The confidence interval is 6.
Join me in a small experiment: take any die you believe to be unbiased. Now, roll it 36 times and record each time each number comes up. Doing it? Okay. We’re going to analyse the data twice: once according to the principles of probability (that you learned in school) and once according to Probability Theory.
Did you get each number 6 times? No. Does that lead you to believe that your die is biased? No. You accept that there is a difference between the theory and the practice, and that in theory the probability is still 1 in 6 for each six numbers. You wouldn’t believe your die was biased unless one number came out sixteen times or more. This is you doing something called “significance testing”. There is an accurate way to do it, but you can kind of guess. Yes the numbers came out different numbers of times, but you don’t feel that difference is significant. So you continue guessing around 1/6 probability.
Now analyse your data backwards. Forget that each face of a die has a 1 in 6 chance of coming up, and look at your data. Using Probability Theory, now tell me whether your die is biased or not? Probability Theory takes the practical results, and builds the principle out of them, which is the reverse of what we just did. The method for doing this is pretty simple, write the following fraction:
(the number of times a number came up)/(the number of events)
I told you to do this 36 times, so it is:
(the number of times a number came up)/36
I know you didn’t do the experiment, so here are my results:
1: 4/36 =1/9
2: 6/36 =1/6
4: 8/36 =2/9
6: 6/36 =1/6
According to Probability Theory my die is biased against the number 1 coming up and for the number 4.