im having some confusion with two formulas, i hope someone can clarify for me

1. the breakeven formula is risk / risk + reward

but sometimes when facing a bet otr i see the formula being used as

2. risk / reward, where reward is current pot + bet faced

when to use each formula?

here's an example i found where they use the 2nd formula but i thought we should be using the first.
thanks very much

I think the biggest issue is the confusion of odds vs. equity. These are very similar concepts, but mathematically they are different.

If you are facing a $50 bet into a $100 pot, you must wager $50 to win $150. You are being offered 3-1 odds. Note that you will get back (assuming no more betting) $200, including your $50 call.
If you do have a 40% chance of winning (another way of saying equity), the odds against you winning are 1.5 -1. Since 3-1 is a better payout than the 1.5-1 needed it will be profitable to call.

Note that I have converted 40% into odds. With 40% to win, we'll assume no chops, and therefore 60% to lose. 60/40= 1.5.

When both are expressed as odds, you have a direct comparison.

If you want to use percentages rather than odds, you need to convert the pot odds calculation into percentages. You must put $50 into a pot of $150, you are contributing 25% of the resulting pot (50/200). Now you can compare the 25% you are contributing to the 40% you believe you have.

Let's suppose you will only win 25% of the time, instead of 40%. Every 4 times you call, you contribute $200 (50*4) to the pot. One time out of 4 you win the whole pot of $200. Therefore you break even. Anything you have above 25% means you will do better than break even, everything below means you will be losing.

Bottom line is that it doesn't matter what formula you use (although I don't they they are explaining it correctly in your example), what matters is understanding the concepts and therefore being able to apply whichever mathematical approach is easier for you. If you need to improve your hand in order to win, you can approximate how many cards (outs) will improve your hand. Let's just say there are 10. If there are 46 unknown cards (10 winners and 36 losers), then your percentage is 10/46 (22%), and your odds against are 36 to 10 (3.6 to 1). It might be easier for you to deal in percentages, and it might be easier to use odds - but the results will be the same. You just don't want to mix and match them in the same comparison.

The formulas you show determines the equity you need to make a call or bet to break-even. Let’s concentrate on Call.

Assume the pot is Pot after an opponent makes a bet of Bet, which you must call.

The reward is your total return if you call the bet = Pot + Call = Pot + Bet

The risk is the amount you had to bet= Bet

So Risk/Reward = Bet/(Pot+Bet) which is your 2nd formula.

The first formula is actually the same IF reward there is defined as Pot, your winnings

To provide support for the formula, after you bet the total amount in the pot is Pot+Bet. Of that amount you contributed Bet (previous contributions are part of the winnings), so your “fair share” is the fraction you contributed = Bet/( Pot+Bet) = risk/reward

The example you quoted is essentially an equivalent way of determining the equity you need to call.

Here's another way. Pot odds = Amount you win (Pot after villain bet)/Call amount

Required equity for breakeven = 1/(1+Pot Odds)

The example you quoted is somewhat misleading IMO. If opponent bets 50 into a 100 pot the pot to you is 150 and you have to call 50 so total return with a win is 200

Risk/Reward = 50/200 = 25%

Another way: Pot odds = 150/50 =3 to 1: Required equity = 1/(1+3)= 25%

Your first formula is the breakeven EQUITY formula. It comes from setting your equity at p, using that equity to write the EV formula, setting EV=0, and solving for p. That is, let r=reward, b=risk (b for “bet”, needed a different letter for risk) and p your equity then

EV = rp - b(1-p). If EV is zero then rp - b + bp = 0, so rp+bp = b, giving b/(r+b), which is formula 1.

Formula 2 is the breakeven ODDS formula. Given an equity p, the equivalent odds in favor of that outcome are p/(1-p). For the equity formula above 1-p equals r/(r+b) so dividing b/(r+b) by r/(r+b) gives b/r, which is formula 2.

The two formulas are equivalent ways to express the same thing. It all depends whether you would rather work with odds or equity. For your example b=50, r=150. Formula 1 gives 0.25 or 25% as your breakeven equity. Formula 2 gives 1/3, or 3:1 odds against winning. If you use the rule of 2 (or 4 ott) to calculate equity, then formula 1 is the correct one to use. If you count outs and get odds (for example 10 outs otr implies 36 losing cards so your odds against are 3.6:1) use formula 2. (3.6:1 is worse than 3:1 breakeven in your example so it is a fold)