Pascal's Triangle

 The above is a simple and brilliant mathematical construction, Pascal's triangle, down to the 6th row. 

Before you count the rows yourself and tell me I'm wrong, there are 7, let me make explicit what the small type on the left hand side suggests. The top "row" consisting of just one call, is the 0th row, not the first. The 1st row has two cells. 

Now let us look at the structure of the pyramid. Every outside cell is "1." Every inside cell derives its number from looking over its shoulder, at two of the contiguous cells, and by summing them. For example, the middle cell in the 2d row says "2." That number is the sum of each of the two cells above it, both of which are contiguous with it and above.   

The middle cell of the 4th row says "6." This is the sum of the "3" over each of its shoulders. 

Knowing this, we can figure out how to continue the pyramid to the next, 7th, row if we want. That row would be, 

                                                     1 - 7 - 21 - 35 - 35 - 21 - 1.

Now: what is the significance of this structure? It tells you at a glance the likelihood of any outcome after you have flipped any given number of coins. 

We can arbitrarily stipulate that the left hand side of the pyramid represents ALL TAILS and the right hand side represents ALL HEADS in any given row, and that the interior cells represent all intermediate possibilities.  So, for example, on the 6th, representing 6 coin tosses, there is 1 outcome in which all flips come up tails. There are 6 outcomes in which there is 1 head and 5 tails; 15  in which there are 2 heads and 4 tails, and so forth across the row. 

To assign a fractional probability to an event, take the numerator for that fraction right out of the relevant cell in this pyramid. Add up all the cells in that row and use the sum of them all as the denominator. 

What is the probability of a 3T/3H outcome of 6 throws? To know this, we go of course to the bottom row of the above version of the pyramid. We go to the central cell on that line. The number there is 20. So that is our numerator. Then we add up all the cells. The number we get by doing this is 64. Simpler for most of us, just intuitively keep doubling as your eyes move down the pyramid: one, two, four, eight, sixteen, thirty-two, sixty-four.  

So ... our fraction representing the 3T/3H outcome is 20/64. Reduce that to 5/16. Voila! IYou know that you have 5 chances in 16 of getting that outcome. If you prefer decimal form, that is 0.3125. Or 31.25% if you please. 

I have no real point to make here. I just enjoy the beauty and the simplicity of the pyramid. Also, it was where my train of associations chugged off to after the discussion of lotteries a few days ago.