A23X on a 78Y board = 21 nut outs

21 outs / 45 unseen cards (52 - 4 in your hand - 3 on the flop) = 46.7%

100 - 46.7 = 53.3 / 46.7 = 1.14:1 ratio

is it the same formula on the turn minus 1 unseen card?

no.

i don't think u know what equity is, because if i'm to understand what you are attempting then you think it is the probability that something occurs and consequently you think 'low equity' is how often a low will occur.

the use of x and y in the specification of hand and board is weird or misleading. if they are meant to be a high cards (any card of a rank 9-k) then specify that because if they are low cards things change.

i'm at a loss to explain why you made the ratio. (although it is the correct ratio of not making a low on the turn to making a low on the turn)



if you are in fact trying to determine the probability that A23J with a flop of 78Q as an example will make a low on the turn you did arrive at the correct probability of 46.7% ott without regard for an opponents range. However, if you wanted to know the probability that A23J with a flop of 78Q will make a low by the river (at showdown) you have to consider 2 streets. To consider 2 streets you have to think in 2 card combos. 45*44/2 =990 is the number of turn/river combos possible. (again this is without consideration for an opponents range)combinatorically this is written C(45,2). in this instance as you need but 1 low card A-6 to make a low, it would seem easiest to determine how many of the 2 card combos (turn/river combos) are both cards 7-k.
determine the probability that you don't make a low and the probability that you do is this values compliment. 24*23/2 =276 are turn/river combos where A23J doesn't make a low, therefore 714 is the number of combos you do make a low. 714/990 =72.12% is the probability of a low by showdwon.

Equity is short for average pot equity at showdown. it is the A23J's expected share of the pot at showdown. it takes into consideration every possible showdown: determines the share of the pot for each one, sums them all and then divides by the total number of possible (hence an average). Equity is not really a low or high thing. when you win the entire pot equity =1, when you win half the pot equity =.5 and when you win 1/4 of the pot equity =.25. in this sense if you knew for certain someone had a high that you could never better, and so the best you could do was win half the pot with a low and consequently .5 equity, then if for example you had a 72% probability of making the nut low then if you were never getting 1/4'd your equity from low would be 36%. if 20% was the probability of getting 1/4'd and hence 52% was the profitability of getting 1/2 then equity from low would be 27%.

I don't think you're using the word equity properly, but I don't think that's what you're asking either.

I think you want to know the probability of making the nut low (a) on the turn and (b) by the river.

(a) is easy to calculate and you've basically done it. There are 45 unseen cards; of those, 21 will give you the nut low on the turn. 21/45 = 46.7%.

(b) is a little bit more complicated. To figure the probability of making a low (not necessarily the nut low but any low) by the river, we have to figure the probability of not making a low and then subtract that from 100%.

On the turn there are 24/45 unseen high or pairing cards (when I say pairing cards here, I mean the remaining three 7's and the remaining three 8's). After one of those high or pairing cards comes on the turn, there are 23/44 unseen high or pairing cards on the river.

To calculate the probability of a high or pairing card coming on both the turn and the river, we multiply these two probabilities:
24/45 x 23/44 = 53.3% x 52.3% = 27.9%

That's the probability of not making a low by the river. So the probability of making a low by the river is that probability subtracted from 100%:
100% – 27.9% = 72.1%.

However, sometimes you will get double-counterfeited, and your low will not be the nut low at showdown. In order to get double-counterfeited, 9/45 unseen pairing cards (when I say pairing cards here, I mean the nine low cards that pair either the A, 2, or 3 in your hand) have to come on the turn, and 6/44 unseen pairing cards have to come on the river. Here's the probability of that happening:
9/45 x 6/44 = 20% x 13.6% = 2.7%

We then take the probability of making any low by the river and subtract the probability of making a non-nut low to figure the probability of making the nut low by the river:
72.1% – 2.7% = 69.4%

If you want to express this as odds, you convert it to a ratio as follows:
100% – 69.4% = 30.6%
69.4 to 30.6 = 2.27 to 1

(This article explains the difference between probability and odds, in case that's part of your question: http://senseaboutscienceusa.org/know...d-probability/.)

To calculate your equity in the hand, we'd need to know how many opponents you have and what their exact cards are (including suits) so we could calculate each player's probability of making the winning high hand and/or the winning low hand.

For example, you could make the nut low but win only one-quarter or one-sixth of the pot—that would reduce your equity considerably. Conversely, you could make the nut low and also make the nut flush—that would increase your equity considerably.