mathematical pardoxes

[kopend]

what is your favourite mathematical paradox.

[Brugel]

The best paradoxes tend to be from either logic or probability. One of my favourites is the one about the crows. We want to see whether it's true that all crows are black. Since the statement "All crows are black" is logically equivalent to "All non-black things are not crows" then we can provide evidence for the first statement by observing lots of objects which are not black and checking that they are also not crows (since if two statements are equivalent then evidence for one must also be evidence for the other).

So if I see a white cat then this supports my claim. So does a red bus. And a yellow submarine. And so on. I can thus provide evidence for the claim that all crows are black without ever looking at a crow at all.

[Brugel]

One from probability. There is a certain virus which affects 0.01% of the population. A test is available to find out whether or not you have this virus but it is not entirely accurate. If someone has the virus the test will correctly indicate this 99% of the time. And if someone doesn't have the virus the test will also correctly indicate this 99% of the time.

Suppose then you take the test and it indicates that you have the virus. What is the chance that you actually do have it (i.e. that the test has got it right)?

The answer, astonishingly, is less than 1%. In other words, even though there's a sense in which the test is right 99% of the time it will still give the wrong answer for 99 out of every 100 people it indicates as having the virus. 

[donslot]

The gambler's paradox...

It has long been accepted that there is an equal chance on the toss of a coin that the result will be heads or tails. But at some stage given enough time and with a large enough sample of tosses and the right kinetics of the spin that surely there is a likelihood of neither heads nor tails (milled edge or no).