Odds, Schmodds.

I've tried to avoid large amounts of arithmetic, but at this point it seems inescapable. The "odds" math I was talking about earlier can be expressed a little more formally.

Odds = (probability of an event)/(probability of non-event)

if we call probability P then odds becomes:

Odds = P/(1-P)

And therefore

P = Odds/(1 + Odds)

Let's revisit an earlier example where we rolled a die without knowing how many sides it has. Our earlier comparison looked like this:

Probability of rolling a six assuming a six-sided die: 1/6

Probability of rolling a six assuming a twenty-sided die: 1/20

or odds of about 3.333 to one.

But what do odds of about 3.333 to one mean, really? Perhaps if we could express this in terms of a probability, like 20%, 50%, of 90% we'd feel like we understood it better.

We already know the odds, we want to know the probability. Thanks to the awesome power of algebra, we can compute the probability from the odds. Our calculations go like this:

3.3333/(1+3.3333) = 3.3333/4.3333 = 0.7692

Therefore, odds of 3.333 to one mean a probability of about 77% that we have a six sided die.

Let's look at our earlier examples and convert them from odds to probabilities.

The "no DNA in the duke rape case" example: odds of 1.3333 to one: 57.1%

A standard rape case with one trillion to one odds: 99.9999999999%

Our trick die that rolled six sixes in a row: 99.9978%

Our example of rolling numbers six our less six times in a row on a six sided die versus a twenty sided die: 97.4%

This also gives us handy guide for thinking about odds and probability. If someone says the odds are 10 to one, we're talking roughly 91%. If we switch to 20 to one, we're talking about roughly 95%. If someone says 100 to one, 99%. One thousand to one, 99.9%, ten thousand to one, 99.99%, and so on.