When I was a child, my grandfather blew my mind with a bit of basic probability and statistics. If you play the lottery, he explained, you might as well just play the numbers 1, 2, 3, 4, 5, 6. Your chance of winning will be exactly the same as with any other particular combination of numbers. I was astonished and incredulous at first, but then I thought it through, and had to admit, of course, that he was right.
Now, years later, I am not so sure anymore.
Of course, the probability argument still holds. The sequence 1, 2, 3, 4, 5, 6 looks decidedly “un-random” to us, but it is exactly as likely to come up in a random lottery draw as the sequence 1, 22, 30, 34, 57, 58, for instance. Of course it is. Every number in the pool has the same chance of being drawn in a single round, and the rounds are independent of each other. My grandfather, who had little formal education, understood this. I understood this as a 7-year old (OK, after my grandfather had explained it to me).
But there is something else.
Here is a first question: What if many people know what my grandfather knows and just play 1, 2, 3, 4, 5, 6, because it’s as good a combination as any? This makes it, by definition, a worse combination than most, because if it really does come up one day, the winnings will have to be shared among more people.
I don’t know how likely this is. My gut feeling is that people who know enough probability to know that they might as well play 1, 2, 3, 4, 5, 6 know enough probability not to play the lottery in the first place. But who knows, there might be a few. What are their chances of winning? I would argue: zero. Not “virtually zero” or “very slim”, but zero.
Here is what would happen in the unlikely event. At first, the draw would proceed as normal. 1, 2. That can happen. 3 would elicit a bit of surprise, maybe a slight chuckle from the lottery draw lady (to use the technical term). At 4, you would start seeing the hashtag #whatsupwiththelottery appearing, and by 5 and 6, it would start to trend. Then, there would be discussion. People would start complaining about fraud or manipulation. Others would try to set them right. A sort of kind of war would erupt on some corners of the internet (#lotteryfraud vs #itssimplestatsyouidiots). Don’t believe it? Listen to this segment from This American Life.
The next morning’s breakfast TV show would feature some sleep-deprived statistics professor trying to explain in simple terms that this combination of numbers is indeed as likely to happen as any other. The breakfast TV presenters would thank her with a half-vapid, half-patronising comment and then move on to the next thing. (“Wow. Thank you very much Professor. I still think it’s weird though, don’t you, Sarah?” – “Yes I do Roger, but then all this maths has always been way over my head.” – [Both chuckle] – “In other news, a puppy had to be rescued from a roller coaster in Bedfordshire …”)
People would carry on agitating though. Maybe a minority, but a vociferous minority would insist that the draw was manipulated and would have to be repeated. They would be completely immune to any sort of scientific or logical argument. They would refuse to consider the evidence or give the matter any kind of deep thought. And they would gain support from a larger number of people who have always been sort of kind of suspicious of statisticians, and of scientists in general. (Don’t believe it? Look at what is happening with the debates around evolution, climate change, vaccines or homeopathy.)
In the end, the public pressure would get so big that the lottery agency would have to give in. There would be a press conference. They would have to annul the draw, and repeat it some other time. Order would be restored. Around the country, a handful of people who would otherwise have won would grumble that this isn’t the statistically correct way to proceed. Their friends would point out that if they knew so much about statistics, they should have known better than to play in the first place. Everything would get back to normal.
And somewhere on a cloud, my grandfather would sit, smile, and slowly shake his head.