Dealing with infinity
Thanks for the responses I thought I may post a follow up to dealing with infinity. It's something that's not entirely intuitive and has a lot of weird aspects. Which I hope I haven't mussed up in the post. :P
In particular something can have a probability of exactly zero and still happen and something can have a probability of one and not always happen (I have heard this referred to as when something happens 'infinitely often'). These sorts of things occur since the probability of something happening is defined in terms of what happens should you, in theory, conduct a test an infinite number of times. The probability is simply the number that is converged to. It then follows that if something happens only a finite number of times and then never again that the probability of it happening overall is zero, not just approaches zero but actually is zero. It is true that something that is impossible has probability zero. The converse, that if something has probability zero then it is impossible, however is not true.
Here are a few examples.
Say you had all real numbers from 0 to 1 in a line and dropped a pin on them so they hit one randomly assuming that any of the real numbers have an equal chance of being hit. Then if you ask the question 'will I hit a rational or irrational number?' the answer it turns out is that with probability 1 you will hit an irrational number. This at first glance seems very counter intuitive since it is realatively easy to convince yourself that between any two rational numbers there is an infinite number of irrational numbers and vice versa. However, it turns out that the rational numbers are countably infinite whereas the irrational numbers are uncountably infinite (yes maths has different sizes of infinite, horrifying I know). Uncountably infinite is so much bigger than countably infinite that, well the result follows.
There is a similar result in computability theory that states if you have a particular problem and out of all the algorithms (description of the steps required to take to find an answer) that solve it I pick one totally at random and give it to you then with probability 1 there will be an algorithm that solves the given problem more quickly.
In a similar vein but not dealing with probability is the proof that 0.9999... = 1. There is more than one way to prove this but the easiest (read most basic) is as follows:
1/3 = 0.33333....
3*(1/3) = 3*(0.3333....)
3/3 = 0.99999....
1 = 0.9999...
It is important to realise that the right hand side isn't merely infinitely close to the left hand side, ie 1, it is actually exactly equal.
To any statisticians reading this I apologise. I have a sneaking suspicion I've conflated expected value and probability. :P
To the placebo comment actually affecting changes in the brain your point is well taken. I was caught up thinking of whether a medication works or not, where placebo is nothing more than allowing a control group. Certainly the merits of placebo medication are a fine debatable topic. ;)