Apr 02, 2013— read in full
Maths problems almost everyone gets wrong
The Monty Hall Problem
This famous problem is named after a game show host, because it takes place on an imaginary game show.
These are the rules. You are shown three boxes: two of them are empty and one has the grand prize in it. You pick a box, and Monty - who knows which box has the prize in it - opens one of the other boxes to show you it is empty.
He then makes you an offer: you can keep the box you chose, or you can swap to the one Monty didn’t open. What should you do?
Most people say it’s a 50-50 chance, because there are two boxes left and one of them has the prize in it - but in fact, you should always switch. If you stick with the box you chose, you’ll only have a 1 in 3 chance of winning.
Here’s why. You have a 1 in 3 chance of picking the right box to begin with, because there are three boxes and only one has the prize in it. If you picked the winning box and you choose to swap, you’ve just given your prize away.
What if you picked an empty box? Monty will open the other empty box, leaving the one with the prize - so if you swap, you win.
Swapping is basically betting that you picked an empty box to start with - and there’s a 2 in 3 chance you’ll be right. If you still don’t believe it, try playing the game for real with some friends, with one person always swapping and one always sticking - you’ll soon see who wins more often.
How big is 0.999… ?
Think about the number 0.999 recurring - that’s 0.9 followed by a string of 9s that goes on forever. It seems like it must be smaller than 1, even if only by a tiny amount - but actually, it’s exactly the same.
If that sounds wrong, don’t worry. Researchers have asked university maths students and maths teachers and most of them got it wrong too. But there are a few simple ways to see that it’s true.
For example, you probably already know that ? is 0.333 recurring. Multiply by 3 and all those 3s become 9s, giving 0.999 recurring. And of course, 3 times 1/3 = 1.
Or you can use algebra:
Let's start by saying that x = 0.999...
That means that 10x = 9.999... - when you times by ten, you just move the decimal point one place to the right.
9x is the same as 10x minus x. So:
9x = 9.999... - 0.999...
In both 9.999... and 0.999..., the 9s after the decimal point go on forever. So when you subtract one from the other, you can simply get rid of all of them. Which means that 9x = 9.
If 9x = 9, then 1x = 1.
We started by saying that x = 0.999... and we've ended up working out that x = 1.
The only way to make that work is if 0.999... = 1
So don’t waste your time trying to write out an infinite number of 9s: writing 1 is much quicker.
The Monty Hall problem shows that it’s easy to get confused by probability. But it doesn’t take the made-up rules of a gameshow to make it happen.
Imagine that there is a disease that 0.1% of people have, and that someone has invented a test for this disease. This test isn't perfect. If you have the disease, the test will spot it 99% of the time, but 1% of the time it will be wrong and say that you don't have it.
If you don’t have the disease, the test will get it right and tell you that you don’t have it 99% of the time, but 1% of the time it will get it wrong and say you do have the disease.
It sounds like a pretty good test: after all, it gets the right answer 99% of the time. The question is: if you take the test and it tells you that you have the disease, should you be worried? How likely is it that you have the disease, rather than the test being wrong?
It might seem like the answer should be 99%. But think about how many people have the disease compared to how likely the test is to make a mistake.
If you test 1000 people, you’d only expect 1 of them to have the disease - that’s 0.1% of 1000. But 1% of the time, the test tells people who don't have the disease that they do have it. That means that about 10 of the the 999 healthy people will be told they have the disease when they don’t.
So if the test comes back with bad news, it’s actually ten times more likely that you’re healthy than that you don't have the disease than that you do - not such bad news after all.