Jun 20, 2011
— read in fullMaths problems almost everyone gets wrong
We’ve all struggled with a maths problem we just couldn’t make sense of. But these deceptively simple teasers have fooled even the sharpest mathematicians.
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 is which - 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 - but in fact, you should always switch. If you stick with your original box 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 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.
Or you can use algebra:
x = 0.999… 10x = 9.999… 9x = 9.999… - 0.999… = 9 x = 9 ÷ 9 x = 1
So don’t waste your time trying to write out an infinite number of 9s: writing 1 is much quicker.
Spotting diseases
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 a disease that 0.1% of people have. There’s a test for this disease that spots it 99% of the time and misses it 1% of the time. If you don’t have the disease, the test will tell you you don’t 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 the test tells you that you have the disease, what are the chances that you really do?
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 999 healthy people - about 10 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 have the disease - not such bad news after all.
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