Can we trust the lottery machine?

This news from South Africa about a set of consective numbers that came out is just too good for a mathematics and ToK teacher. It is just so charged with potential for investigation and discussion so I needed to write a quick blog with some associated thoughts while they are on my mind. Read the article, but essentially, the numbers 5, 6, 7, 8, 9 and 10 and not in that order I might add,  came out of the lottery draw and has played on the human psyches' sense of scepticism based on an intuitive idea that it is surely too unlikely to be possible...

What are the issues and questions?

Reaction - What I like here is the idea that those with experience in the idea will be the first to point out to everyone that the these 6 numbers are just as likely to come out as any other given set of 6 numbers. Others without experience will have a strong sense of doubt because it is a counter intuitive. For me though, it is important to consider some middle ground here....

How likely - To explore this we have to a lot of asking 'how likely is...?' and the associated 'how manys?'

• In a traditional 6 ball lottery (50 balls) - how many possible combinations are there?
• They are all equally likely, but how many of them will be a set of 6 consecutive numbers?
• As such, what is the likelihood of drawing 6 consecutive numbers?
• This is then related to 'how often would expect this to happen?' 

Unlikely events - The above yields that of the many millions of possible combinations only a very small number are consecutive numbers. As such a group of 6 non consecutive numbers is much, much more likely. This is why we are drawn to this result and why it appears to stand out. The thing with unlikely events though is that if you do something often enough, the unlikely event becomes very likely to happen, at some point. So we shouldn't be surprised if and when it does. 

How many draws have there been ? - So the question is how many lottery draws would there have to be, before I could be pretty sure to have had a set of consecutive numbers? The, have we had enough? Although, of course, something that might happen 1 in 300000 times could be the first thing to happen. Still, its a good exercise to find out what sort of prevalence you could reasonably expect. If something you would expect to happen 1 in 300000 trials happened every second time, you would have reason to 'probe'

20 Winners and why - This bit is great! The fact that there were 20 winners is another reason for people to be suspicious, but now you have to get in to the idea that it might be more likely for people to pick 6 consecutive numbers than any other set of 6 numbers. You might know that 5,6,7,8,9,10 is just as likely as any other to come out and fgo with it. Still, if you knew that, you might also know that  you are less likely to be alone....

Evidence that warrants probing - So, the big question now is, when might I have the kind of evidence to justify a 'probe'? What could happen that is Soooo unlikely that I might begin to suspect foul play? This is a great question. We can all see extremes like the one suggested above. If a 1 in 300000 event happened 50 times in 100 trials I would be suspicious.., but where is the line. Of course we know it could happen, but....

How is related to statistical significance? - And this might be a nice way to introduce the idea of statistical significance and samples that comes in to hypothesis testing, where we effectively say of some results that whilst we know they 'could' happen, they are quite unlikely after a certain point.

Lottery variations - A quick look in to this particular lottery will tell us that it is a 5 ball lottery (numbers from 1 to 50) with a power ball that comes from 1 to 20. That, of course, changes things and makes for a slightly more complex calculation.

Exploration? - So now I am thinking about potential here for exploration and it seems to me that there is a lot. A quick google search on the history of lottery results reveals that there are lots of databases of previous lottery draws that could be used for exploration. Theoretical probability analysis can be used to work out the probabilities in different types of lotteries of getting any given combination of 6 numbers. Then you can work out the probability of any of them being consecutive numbers and work out the corresponding prevalence you would expect etc etc etc as I have suggested above. This could take you a long way.

Analysis of the results could lead us to exploring questions about distribution, prevalence and sampling. What about introducing the notion of order? What are the chances of the 6 balls coming out in size order? It seems to me to have lots of potential, and I am now challenging myself to map out what an exploration would look like..... I'll update the blog post accordingly

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