## Essential Basic Algebra Transformations, and Probability in terms of Odds

Okay…here’s a type of algebra transformation that can be counterintuitive if you get stuck at a certain point but is very important!

Say you have odds=p/(1-p), where p=probability. How do you solve for probability in terms of odds? You might start with multiplying both sides by (1-p) which leaves you with odds(1-p)=p. You might think, rats, that’s still too many p’s everywhere! If you multiply that out, you get odds-odds(p)=p. Rats, still too many p’s! But you can get rid of some of the p’s by dividing both sides by p–you’re trying to get rid of as many p’s as possible. So dividing both sides by p, you get (odds/p)-odds=1. Yay! Don’t worry that you got rid of the p on the right side, you still have one on the left side. Now you only have one p: (odds/p)=1+odds.

Then flip that over, p/odds=1/(1+odds). Then p=odds/(1+odds).

Then you can simplify that even more, if you want! You can divide both the top and bottom of the right side by odds (the same as multiplying both the top and bottom by odds^-1). The top, odds/odds=1. The bottom, (1+odds)/odds=(odds^-1 +1). So that’s p=1/(odds^-1 +1).