What are the odds? (Poker Question)?

Question

This is a bet I have with a friend and we can not get a final answer to determine a winner, so any help would be appreciated.

Here is the scenario:
In Texas Hold’em Poker, with 1 52 card deck in play the following has occurred.
There are 3 players, and 4 community cards down, 2 cards have been “burnt” and each player has 2 cards, leaving 40 cards in the deck.
The community cards are 3 3’s and an Ace.
One of the players has an Ace and an 8.
So the question is: “What are the odds that one of the other players has the other 3 in their hand? (they each have 2 cards remember for a total of 4 cards)”
The Odds of the 3 3’s being down is not in question as we all agree that it is high. The only question is the odds of the 4th 3 being in someone else’s hand.

Answer 1

There are all kinds of flaws in the answer above.

First, if another player has an ace, his kicker is insignificant as they would chop the pot with 3's full of aces.

Second, there are 6 cards which the player with A8 knows (333AA8) so that leaves 46 unknowns. Therefor the odds of one of the other players having that 3 is (1/46)4 or about 8.7%.

Third, we cannot assume that this means as a player we have an 89% of chance of winning. Aside from the chop pot scenario, a player with any pocket pair higher than 22 could outdraw us on the river and make a better full house. Also, if the suite of ace matches any of the 3's on the board, 25, 45, or 24 suited could make straight flushes.

Answer 2

It depends on whether the burned cards were exposed or not. If they were not, they're insignificant.

In reality, the player with the ace doesn't know where 47 of the cards are. The odds that the other player has the final 3 are 1/47 + 1/46. If the burn cards were exposed, that increases to 1/45 +1/44.

However, that other player may also have another ace, the odds of which are 3/47 + 3/46 (or 3/45 +3/44). If he does have an ace, then it is a question of whether he has a 9 or better.

Out of the remaining cards, 20 are above an eight, or just about half.

I would estimate the chances of the fellow with the ace winning at 1-((1/47+1/46)+((3/47+3/46) x .5)) which is about 89%.

Answer 3

well first of all you can't count the burn cards as being out of the deck because no one knows what they are, they're unseen

now, 1 out of the 42 cards remaining is a 3, so if there are 4 cards between the remaining 2 players then the odds should be 42/4, or 10.5/1 against the fourth 3 being in someone's hand

Answer 4

The question your asking is simply "What are the odds of a 3 being on one of the other 2 peoples hands" which is only 4 cards. So, what is the odds, of the 4 cards dealt to those 2 people, being a 3.

The odds of a 3 being dealt, from a 52 card deck is about 7.6%, then from a 51 card deck is about 7.8%, then a 3 from a 50 card deck is 8%, then a 3 from a 49 card deck is about 8.2%(all those are rounded)

So, average those, and you get about 7.8% chance that in the 4 cards you don't know that one of them is a 3.