I have been wondering if there is anyway to calculate the probability of making your hand with two cards to come without working backwards? I see the most common suggestion is to find the probability of not making your hand and go from there.

For example, lets say I have 9 outs with two cards to come, how do I find the probability of making my hand? I know that I can find (38/47) and (37/46) and multiply those two together and find the probability that I won't make my hand, but why doesn't this principal work in reverse? If I multiple (9/47) by (9/46) I end up getting something like 4 percent. Is there a way to find the probability of making my hand without working backwards? Obviously this isn't very detrimental to my poker success because I can find the odds by working backwards, but this is just something that I have been racking my brain about for the past two days. Thanks for the help!

Doing it the complementary way is just easier and more understandable to many people. For the more direct approach, one formula is:

Pr(Win) = Pr(Win on turn) + Pr(Don't win on Turn)*Pr(Win on River given don't win on turn)

For your example:

Pr(Win) = 9/47 + (38/47)*(9/46)

The formula 9/47 * 9/46 makes no sense. It is equivalent to the probability of winning on the turn and winning on the river given you did not win on the turn???

For those interested in the math,
9/47 = 1-38/47
9/46 = 1-37/46
So 9/47 + (38/47)*(9/46) =1 - 38/47 + 38/47*(1 - 37/46) = 1 - 38/47 * 37/46

Not sure of this is what you meant but....

I read in a book somewhere this easy way to do it in your head:

If there's 2 cards to come you multiply your number of outs by 4.
In this case 9*4=36%

If there's one card to come you multiply by 2.
9*2=18%

These are obv approximations but can give you a general idea.
Can anyone confirm this?
Sorry if I'm completely off here, I'm a total nit.

Yes that's correct, although there is a little improvement for that formula, 2 cards to come outs x 4 -1, 2 cards to come outs x 2 +1

ye it gives a good approximation, but is inaccurate sometimes. for example, if you have a flush draw + 2 overs otf (9 outs to flush + 6 outs to pair = 15 outs).
15*4= 60. although your equity will be good, it will not be this good.

i think the 'rule of 4 and 2' can sometimes overestimate big draws.

It does, but big draws appear somewhat infrequently and when they do, you instinctively know that it's shove time, because you'll be a favourite. There, you don't need the rule of 4 and 2.

Thank you! For some reason I didn't realize that I was attempting to calculate my probability of winning on the turn and the river. This perfectly answered the question, thanks again.

Right. A useful adjustment for the rule of 4: apply the rule, then if outs greater than 8, subtract (outs - 8) from the percentage. E.g., for 15 outs with two cards to come: apply the rule of 4 to get 60%; then subtract (15 - 8) or 7 from 60 to yield 53%. The actual probability is 54%.

For 12 outs, such as flush draw plus gut shot: rule of 4 gives 48%; adjustment gives 44%; actual is 45%.

I think the 2x and 4x rule by itself is enough. the idea is to get approximate odds. There's almost know value in knowing the exact odds to the tenth of a percent. From there your brain power is better devoted to thinking about how the hand played out, notice chips stacks and player types.

If you are spending all your brain energy trying to determin if you are a 43%er or a 46%er then you are going to make mistakes because you are going to miss the important details of the hand.

The idea of doing a more accurate calc to the nearest percent is that eventually you will memorize the more accurate number and not have to do the calc anymore. If you use the rule of 2 and 4, your memorization will consist of the inaccurate numbers. Why would you want to do that?

If you need to expend 'all your brain energy' to perform simple subtractions like the ones Steve Brecher described above, then you're probably already making all kinds of mistakes.

I agree completely that most of the time, the rules of 4 and 2 are perfectly good approximations. But there can be situations where the equity difference is quite important.