Confused about pot odds

Question

I wanted to practice my implied and pot odds calculating skills and found this website online http://www.philnolimits.com/uploads/8/1/8/9/8189328/practice_pot_odds_and_expected_value_problems.pdf And it had this problem which I think was wrong:

All In? You have $50 remaining in a cash game and you flop a small flush draw. There is
$30 in the pot and your opponent goes all in. Should you call with your flush draw?
The only way you can win is to hit the flush on the turn or the river.

The answer was right but the thought process looked flawed. It goes like: First, use the Rule of Four. You have 9 outs with two cards to come. 9 x 4 = 36%.
You have about a 36% chance of making your flush and winning the pot. $30 (pot) + $50 (his bet) + $50 (your call) = $130 total pot. You have to call $50 to win $130. $50 (your call) / $130 (total pot after your call) = 38% So, by a slim margin, you should fold.

I mean you shouldn't take into consideration your 50$ bet into the amount you win. I think it should be like 30$ + 50$ = 80$ pot and you have to bet 50$, so your getting 8 to 5 odds to call there. Am wrong or is this article flawed?

Answer 1

Here is another way to look at it:

Gained if you call and win: 30+50 = 80
Lost if you call and lose: 50
Your Equity = 0.36
EV = Equity(Gained when win) + (1-Equity)(Lost when lose)
EV = 0.36(80) + (1-.36)(50)
EV = 28.8 + -32
EV = -3.2

Plug in 0.38 for your equity and you will see that its indeed near a break-even call.

How did they get that formula?

Lets say:
E = Your Equity
P = Pot
B = Bet that your facing

Just like above...
EV = E(P + B) + (1-E)(-B)

Lets set EV to 0 (since we want to find out where the b/e call is) and solve for E  :
0 = E(P + B) + (1-E)(-B)
0 = EP + EB -B + EB
B = EP + EB + EB
B = E(P + B + B)

B / (P + B + B) = E
E = B / (P + B + B)
E = 50 / (30 + 50 + 50)

Answer 2

It's a common misunderstanding. You can do it both ways, it's up to what's more convenient to you.

European Way (include the amount you bet)

equity (%) = your_money/(pot+your_money) = 50 / 130 = 38%

American Way (not include your bet)

equity (us) = your_money/pot = 50 / 80 = 5:8
==> meaning you will win 5 times and lose 8 = 38% (European)

The flush draw probability if you go all in from the flop is in European odds = 36%,
but in US odds ~ 1:2.

The catch is to be consistent, so either calculate everything is US odds or European,up to you really ;)

EDIT: Lessharm has given the mathematical approach, which will clarify it more

Answer 3

When you go all-in there is no implied odds

On the flop if you go all in then you are not going to face another bet so you can go with the odds of the next two cards.

Hand odds = (1 - your probability) / (your probability)
You have 9 outs
Hand odds for one card is (45 - 9) / 9 = 4 : 1
But you have two cards to come so your hand odds are
(2/3) / (1/3) = 2 : 1
You can call a pot sized bet
35% is close enough to 1/3

Pot odds
You would be risking 50 to win 80 so your pot odds are 8 : 5 = 1.6 : 1
Your 50 that you get back is not a winning

Pot odds less than hand odds so you should not call
And you could be drawing dead to a bigger flush

Remember you go on the bottom (denominator) in hand and pot odds

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