My Math is Wrong and I Don't Care

I'm not a mathematician. In fact, I would most likely irritate any mathematician worth his or her salt. This is because I am sampling from a mathematical buffet but am far too lazy to clean up after myself or, heaven forbid, prepare any of the buffet items.

A great deal of my thinking is influenced by something called Bayes's Theorem. Bayes's theorem and Bayesian Inference are often described as a guide on how to update or revise beliefs in light of new evidence.

I think that's fair statement of what I hope to accomplish.

I'm not too rigorous with the math, which means my calculations are most likely wrong at some level. I am cleverly selecting examples that I understand to illustrate my point before traveling into uncharted territory. This is much like a child who uses water wings before learning to swim. What is true is that my math is close enough so that my results make sense.

My earlier math appears wrong if you think about it carefully. Remember how we calculated odds?

Odds = P/(1-P)

Yet I was talking about odds of 3.3333 to one when dividing 1/6 by 1/20. The probability of not rolling a six on a six sided die is 5/6, not 1/20. So what gives?

It turns out that when I'm comparing the 1/6 and 1/20 probabilities, there is a Prior Odds hiding in the calculation that makes the math work. When I divide 1/6 by 1/20, I am assuming that I'd be equally likely to select a six or twenty sided die. This implies a prior odds of 1. It's there, we just don't notice it because it does not effect the calculation.

What that means is

Prior odds = P(selecting a six sided die)/P(selecting a non-six sided die)

Which is another way of saying

Prior Odds = P/(1-P)

Since

P = (1-P)

Therefore

Prior Odds = P/(1-P) = 1.

Therefore the only thing affecting the calculation is the DNA evidence.

In the earlier cases, where I merely divide 1/6 by 1/20, I'm assuming a prior odds of 1. Therefore even though 1/6 divided by 1/20 is not an odds calculation, because the prior odds are present the math will work out. Every calculation I've made without a Prior Odds in fact had one. The real calculation was

Odds of a six sided die = Prior Odds of Choosing a Six Sided Die x (Probability of rolling a six with a six sided die) / (Prob. of Rolling a six with a 20 sided die)

or

Odds of a six sided die = 1 x (1/6)/(1/20) = 20/6 = 3.3333