The Bell Curve 1 - to scare off the wimps

This is the equation that describes how something that can happen only one of two ways, p, and q, will happen if you make it happen n times. Say you have ten coins and you flip them all together 1024 times.

The equation says (very briefly) that in an ideal world you will get:

0 heads and 10 tails - 1 time

1 head and 9 tails - 10 times

2 heads and 8 tails - 45 times

3 heads and 7 tails - 120 times

4 heads and 6 tails - 210 times

5 heads and 5 tails - 252 times

6 heads and 4 tails - 210 times

7 heads and 3 tails - 120 times

8 heads and 2 tails - 45 times

9 heads and 1 tail - 10 times

10 heads and 0 tails - 1 time

Or to put it another way: the probability or "chance" of getting 0 heads and 10 tails is 1/1024 or 0.000976563 (plus or minus a smidge; my calculator can display only 10 digits).

Similarly, the probability of

1 head and 9 tails is 10/1024 or 0.0098 (rounded to four decimal places)

2 heads and 8 tails is 45/1024 or 0.0439

3 heads and 7 tails is 120/1024 or 0.1172

4 heads and 6 tails is 210/1024 or 0.2051

5 heads and 5 tails is 252/1024 or 0.2461

6 heads and 4 tails is 210/1024 or 0.2051

7 heads and 3 tails is 120/1024 or 0.1172

8 heads and 2 tails is 45/1024 or 0.0439

9 heads and 1 tail is 10/1024 or 0.0098

10 heads and 0 tails is 1/1024 or 0.0001

These are shown on the "bell-curve" graph.

Enough with the numbers already. There are two important things to especially notice here:

1, the distribution of probabilities is symmetrical.

2, this is for an ideal situation; in a real test the numbers will not come out exactly like this (but that's the way to bet - the tendency will be towards this distribution especially if you flip the ten coins a "very large" number of times. How big "very large" is, we don't have to think about, Deo gratias).

G.K. Chesterton said in Orthodoxy, Chapter 6, "The Paradoxes of Christianity":

"The real trouble with this world of ours is is not that it is an unreasonable world, nor even that it is a reasonable one. The commonest kind of trouble is that it is nearly reasonable, but not quite. Life is not an illogicality; yet it is a trap for logicians. It looks just a little more mathematical and regular than it is; its exactitude is obvious, but its inexactitude is hidden; its wildness lies in wait."

and

"It is this silent swerving from accuracy by an inch that is the uncanny element in everything. It seems a sort of secret treason in the universe."

And Damon Runyon (also well worth reading) said: "The race is not always to the swift, not the battle to the strong, but that's the way to bet."