How can I combine winrate, (risk/reward), and expectancy into a single formula?

External question — Personal Finance Stack Exchange Author: Russel Walker Original post: https://money.stackexchange.com/questions/91039 License: CC BY-SA 3.0 — https://creativecommons.org/licenses/by-sa/3.0/ Adaptation: HTML converted to plain text; contact email addresses removed. winrate = wins / (wins + losses) I fiddled around with it and found that: requiredWinrate = risk / (reward + risk) Expectancy formula expectancy = (winProbability x averageWin) - (lossProbability x averageLoss) So I reasoned that maybe expectancy = (winrate x reward) - ((1-winrate) x risk) Now I have two problems with this formula. One is I cannot be sure that it's correct and two is I cannot figure out how to solve for winrate or how to solve for (reward/risk). In forex trade I may have a potential trade with given reward and risk values and given win rate in which case I would want to calculate expectancy to know if it has positive or negative expectancy. But I would also like to know what winrate I need in for given reward and risk values and positive expectancy in cases where I do not have a known winrate but can make a reasonable decision if I can calculate the required win rate. Edit: I've confirmed that the formula is correct with a python script that simulates trades on a given winrate and compares a calculated expectancy and computed expectancy. https://pastebin.com/8FF1vAMm

External answer — Personal Finance Stack Exchange Author: Russel Walker Original post: https://money.stackexchange.com/a/91060 License: CC BY-SA 3.0 — https://creativecommons.org/licenses/by-sa/3.0/ Adaptation: HTML converted to plain text; contact email addresses removed. winrate = (risk + expc) / (risk + reward)