What is the probability of tossing a fair coin, and it landing on "heads" one, two, five, ten, 50 times in a row? One in two, one in four, one in 32, one in 1024, and one in 1.12x1015. Approximately.
What is the probability of rolling a fair die, and it landing on "1" one, two, five, ten times in a row? One in six, one in thirty-six, one in 7,776, and one in 60,466,176.
This is why the Law of Averages (also known as the Law of Large Numbers) works. It works because the probabilities of such events happening multiple times in a row is virtually impossible. Not completely impossible, no. Some day, every roll of a certain die will turn up "6". So, the world compensates by allowing all other possibilities to happen as well. Funnily enough, they all come out to be about equal.
How many heads am I going to get if I toss a coin ten thousand times? I estimate 5,008. Why? Because it is exceptionally close to the average, and unfortunately, no coin is ever quite fair, as most of them are weighted towards the tails side, therefore more heads will turn up. But only a few.
Try out the law of large numbers yourself. You can either take a day off work and spend a few mind-numbing hours rolling dice, or tossing coins... Or, you can do it a quick (but by no means preferred) way. Set up a spreadsheet, say in MS Excel, and do the following:
- In cell B1, type in
- In cell A1, type in
=RAND(). This generates a random number between 0 and 1, to about fifteen digits or so.
- Note the value now in cell B1.
- AutoFill cell A1 down to cell A5, and note the value now in cell B1.
- Repeat step 4 down to cells A10, A50, A100, A500, A1000, A5000, and, if you're feeling gutsy, cell A10000. Note the value in B1 each time.
Hopefully, you would have found a value between 0 and 1 with only cell A1 filled. By the time you get to cell A50, you would have found a value between 0.45 and 0.55, and by cell A5000, it would be 0.500-something or 0.499-something. I may be out a bit, not having tried this experiment often enough to get typical numbers.
The Law of Averages is at work. Although random numbers on a computer are not, in fact, completely random, they are so close to random that this experiment will provide you with very similar results to that of tossing a coin: namely, obtaining probabilities of a shade over or under 0.5, or one in two.
By the way, if someone throws seven heads in a row, don't be suspicious. Yet. They have one chance in 128 of pulling it off. Only check the coin after about twelve in a row. Why? They now have one chance in 4096. That, for me, is enough to think that someone's lead-plated one half of the coin. The Law of Averages says that they should have thrown between about four and eight tails by now.