WHAT ARE THE ODDS?

"Did you hear about that couple who changed their normal numbers one week?" asked my Dad.

"Oh, and then their old numbers came up?" asked someone - my sister I think it was.

"Yes! And guess what they did the next week? Would you believe it, they changed their numbers back! I mean the chances of those numbers coming up again..."

"Um..." I said, raising a finger. "Aren't the chances... exactly the same as any other set of numbers?"

"What?" said my sister.

"Well, the statistics are the same for any set of numbers," I protested geekily, "the lottery balls don't have a memory!"

This is a weird feature of probability don't you think? I mean, intuitively, if you roll a six, and then another six, and then a third six, you'd think the chances of rolling a six again for the fourth time are miniscule, wouldn't you? But the truth is, as you hold that die for the fourth time, defying the odds in that one instant, you've actually got just as much chance of rolling a six as you had the time before. It always is,  and always will be 1/6 (16.7%)... unless the die somehow knows what's just happened.

It's this feature I think, that makes gambling so appealing - it kind of tricks you into thinking you've got more chance than you really have sometimes. I'm really fascinated by the psychological effect it has on us - a random set of numbers picked for the lottery somehow seems much more likely to come up than a nicely ordered sequence. Don't be fooled though. 1,2,3,4,5,6 is just as good a selection as any other.

This is what makes the Monty Hall Problem work I think, though from slightly the other way around, it proves that probability can be a bit tricky to believe. Monty Hall is a really famous puzzle that plays with your perception of statistics. Forgive me if you've already come across it, but it works like this:

You're a contestant on a game show with a really simple premise. There are three closed doors and behind each of these doors is a prize. The host tells you that behind one of the doors is a car, and behind each of the other two, they've put a goat. You win whatever's behind the door you pick. Now the host knows exactly what each door conceals and waits for you to choose one. When you've selected, the host then opens one of the remaining doors to reveal a goat. The other two doors remain closed and the host asks you whether you would like to stick with your original choice or swap to the other closed door. What should you do? Stick or switch? Does it matter?

Well, the answer, as counterintuitive as it appears, is that you should always switch doors, as it doubles your chance of winning. You might think this is bonkers but it is actually true - and it's true because the probability of a thing happening doesn't change because of an independent thing also happening - just as the probability of a set of lottery numbers coming up this week has nothing at all do with the numbers which came up last week. If it did, you could probably claim that the lottery is not random at all, and is exhibiting all the signs of being either rigged or predictable.

The fact is that when you selected the first door, you had a probability of 1/3 of selecting the car. The host opening another door has no effect at all on that probability, once you've picked - the car and the goats are still in exactly the same configuration as they were moments ago. Therefore, the probability that the door you've chosen is hiding the car... is still 1/3 (33%). This also means that the probability of the car being behind the other door is now 2/3 (67%) as there are no other possibilities.

There was a silence around the coffee table. I think my Dad was trying to figure out a way to prove me wrong and my sister scratched her head as though confused by the whole thing. I was hoping to have a discussion about how compound probability changes when the variables depend on each other - for example, when one lottery ball is chosen by the machine, you've got a better chance of matching the next one - and how it doesn't when the events are independent. In the end, my Mum looked at me and broke the tension with a sentence that always carries a high probability of being spoken in our house.

"Put the kettle on, Jimper," she said.