What are the odds

On a 9 person table - all in pre flop - of hitting an ace (and the ace being the only card that can make you win)

i.e.

Ace 5 vs pocket kings or ace 7 vs Jacks something along those lines

Assuming that no other people on the table were dealt an Ace

Thanks

Well, we can sort of simplify things, I think, by considering all the cards in the unknown deck to be "ace" or "not ace". There are 3 aces cards and 45 non ace cards in the deck. You said no one else has an A so that means that 14 more non-ace cards are in your opponents hands, leaving 31 non-ace cards in the deck.

So the easiest way to do this is to figure out how often none of the cards on the board will be an ace.

This is
28/31*27/30*26/29*25/28*24/27
which is about 57%
so 43% of the time, he'll hit his ace.

Hm twodimes doesn't agree with me for some reason. Did I miss something?
http://twodimes.net/poker/?g=h&b=&d=...+5d%0D%0AKd+Kc

Twodimes doesn't agree because it doesnt exclude the times that an A flops and KK still wins by hitting a set or getting a straight.

Having said that, I can't see why setting it to this configuration doesn't get it to at least over 43%

http://twodimes.net/poker/?g=h&b=&d=...+5d%0D%0AKd+Kc

Yeah you're right, eliminating the other two kings makes a huge difference.

If you're thinking about just how often an ace will come in your scenario then I think my numbers are more or less right. I should have chosen better dead cards for twodimes.

The number of players at the table is irrelevant to the calculation. You know 4 cards, and 48 cards are unknown. You know there are 3 aces among the unknown cards. You want to know the chance that one or more of them will hit the board.

All you do is calculate the chance of no aces coming out on the board and subtract that from 1 (100%). There are 45 non-aces for the first card, then 44 non-aces for the second card, etc. So the calculation is:

1 - ( 45/50 * 44/49 * 43/48 * 42/47 * 41/46) = 42%

Note however, that 42% is NOT your chance to win the hand. The opponent also has outs to improve.

The number of players matters because he stipulated that they do NOT hold an ace, making the deck richer in aces.

In a situation where you have one over such as a7 vs kks or jjs or whatever.
I notice that the preflop odds are 70-30.
On the flop there is a 30% chance of hitting the ace. Why are the odds the same? Because the K still has a chance to hit trips.
I think that if they had j10 and you hit your Ace it would be unlikely that the board would be AJ10 or AJJ or A1010.
It would mean though that on a board of A 10 4. He would have more outs to improve. He has 5 cards to hit and you would like to pair your kicker or ace.
A pairing of your kicker would make his 5 outs become 2 outs. A pairing of his kicker means you would still have 5 outs. If either were to develop trips it would be lights out

No, the dead cards are unknown and you can't make that assumption. If all cards were face up and you knew that 14 more cards were eliminated as non-aces, the calculation (albeit a totally useless one) would be:

1 - ( 31/50 * 30/49 * 29/48 * 28/47 * 27/46) = 92%

That number is meaningless. If your assumption is wrong and 3 players folded Ace-rag, then your chance to hit is actually zero, so it could be anywhere in between. You always count all hole cards as unknown, because it is impossible to know. The 42% answer is the correct one, even though you might guess it to be higher due to players not folding aces. Most players don't call an all-in with ace-rag.

The number is meaningless because you did the calculation wrong. Look at the way I did it. The key is that there 31 non-ace cards and 3 ace-cards in the remaining deck. The odds for the first card coming out a non ace are therefor 28/31, not 31/50

Look, he's asking for something very specific - how often will someone with something like A5 vs KK hit an A, given that none of your opponents has an ace.

You might not think this is applicable to actual play, but it's an answerable question with the parameters given.

also in my example if aces or j10 both didn't improve that would be pretty great.

You're right, but we both did it wrong. Your denominator should be 34. With 34 unknowns, and 3 Aces among them, the first card has a 31/34 chance to be a non-Ace and a 3/34 chance to be an Ace.

1 - ( 31/34 * 30/33 * 29/34 * 28/33 * 27/32) = 39%

And I used 50 for my denominator in the first example instead of 48 (we know villain's cards) so the 42% is off too. My answer should have been:

1 - ( 45/48 * 44/47 * 43/46 * 42/45 * 41/44) = 29%

which is the realistic one when you don't know the folded cards.

Sorry for the bad (quick) math.

Ah you're right, 34 for the first denominator.

We are agreed. 29%

I had said 30% but it was clearly 29%.
Hey guys are there decimals attached to that 29%? Or did you round up from 28%?

You were talking about preflop equity of A7 vs. KK being 70/30 but that 30% is not just because of the chance to hit an Ace. The A7 hand could hit a running straight with one hole card, a running flush with one hole card, hit two 7s for trips, etc. When you add all those up, and add up all the possibilities for the KK to improve too, you end up with the 70/30 preflop equity, which accounts for all possible outcomes. So comparing those two numbers isn't very interesting and they just happen to be pretty close in this case.

But to answer your question, the chance to hit the Ace on the board is exactly 28.65% when you do not know the dead cards.

Comparing to the preflop equity of A7 vs KK, that is an average of 29.6% but
it varies depending on the suits (flush possibilities). If we make it As7s and KcKd then that equity goes up to 32.59%. If we make it As7d and KsKd then it goes down to 28.55%.

In my first post I said that it is 70-30 preflop. It is 30 percent for an ace on the flop. why is it like that? well its because for some reason the other players cards are still very likely to win. if you both flop a pair hes got two shots at 5 outs.

Also we should avg those out to 30%

That is incorrect. It is 30% to be the best hand at the showdown, not to flop an ace. The chance for the A7 hand to flop an ace is about 18% when we know villain doesn't hold one.

what? let me look it up

Flop means the first 3 community cards, not 5 cards.

whats the name of that website everybody goes to for basic odds? I think it has green trim or something. Like a tan backgground. You can find odds for flopping a pair

well i'll be dipped. It said when you have AK the odds of flopping a pair is 2 to 1. I guess half of 33% is like 17.5%

The question was

What is the probability of hitting an Ace on a hold’em board when you have an ace and 8 other players do not.

Presumably, hitting an ace means one or more aces.

18 cards have been dealt. That leaves 34 remaining, of which 31 are not aces. The probability on NOT getting an ace on the 5 card board is then

(31x30x29x28x27)/(34x33x32x32x30) = 0.69.

Therefore the probability of one or more aces on the board is 39%.

Odds are then 61 to 39 against or roughly 3 to 2.

thanks for th replys guys

I dont know how some of you work these figures out (een when you post the calculations lol)

reason i ask was basically a small blind vs big blind i had ace 4 he had pocket kings and i hit the ace, thought he would be extremely p'd off!

but then again if its around 40% i guess its not too bad a beat

Lets see this question first from a flop only point of view.

52-18=34 cards left and 3 aces. Chance you get an ace or more at flop then is = 1-chance you dont get ace at flop. This last chance is 31/34*30/33*29/32=0.7512. Therefore you get at least one ace 24.88% of the time.

In fact the chance you get only 1 ace (no 2 or 3) is less obviously and its equal to
3/34*31/33*30/32*(3!/2!/1!) =23.31%. You get 2 aces 3/34*2/33*31/32*(3!/2!/1!)= 1.55% and 3 aces 3/34*2/33*1/32=0.02% (rounded off to second decimal of %) . They all of course add up to 24.88% .


Moving to a full board all the way to the river with 5 cards dealt the above numbers become;


0 aces ;
31/34*30/33*29/32*28/31*27/30=61.06% hence 38.94% at least an ace

1 ace;
3/34*31/33*30/32*29/31*28/30*(5!/4!)=33.92%

2 aces;

3/34*2/33*31/32*30/31*29/30*(5!/3!/2!)=4.85%

3 aces;

3/34*2/33*1/32*1*1*(5!/3!/2!)=0.17%



Now of course the question is what good is any of this since you cant exactly know that none of the others have received any ace and also clearly ther are many ways one can win without an ace or lose even with 2 aces etc!

And thats why i focused on the flop question first since poker is vastly impacted by action at flop when you have many opponents. Whoever is very strong at flop and only at flop is the most likely to move on. What good is that you get an ace on the river if you are already out at flop from someone that had pocket queens etc.


The more important question is to ask what is the chance that you having a hand like AX will get an ace at flop vs n opponents and have the best kicker too!

And not only that but clearly you may want to add assumptions here like was there preflop raising going on and if so how has this impacted the chance someone would play a hand like AY or not.

Now that is a real decent problem with practical value. Adding more value one can ask what is the chance if you hit that ace that vs n opponents on given ranges each you have the best hand at the time ie the board is A Z W you have AX and you face n opponents. What is the chance your A pair now is the best current hand. You cant really ask what is your equity because a big bet now will eliminate some players that had they been left all the way to the river could still form a better hand in some remote but non negligible ways. So i guess the real question is what is the chance your A pair is the best hand at the time .

Example there is a raiser and 2 other flat callers and you at the big blind have A7. You ask yourself what is the chance that at flop if you now call and see flop , you hit an ace and your hand is the best . Such question is important because with a hand like that you basically need to hit your ace to go any further (ok you can also hit 7 , draws etc) and you need to know if you hit it what is the chance you are leading all at the table and what is the chance you are dominated either by a better kicker AY or by a stronger hand (eg set, 2p etc) in which case you stand to lose a lot (potentially a lot more than what you win if you take the pot down now if nobody else has a better hand to call you-risk reward of playing junky looking AX and how much your flop bet needs to be to balance the risk of domination etc) . Such questions could then impact how you bet , especially if you act first (or they check to you)