flush odds

[cfjr2]

If I have 2 hears in my hand and there are 2 on the board at 4th street, why is it assumed there are 9 outs available?
If, on average, 1 in 4 cards is a heart, and 20 cards were initially dealt to the 10 players, then on average 5 hearts were dealt and I have 2. This leaves 3 dead in various hands and 2 on the table leaving 6 outs (on average - it is of course possible all 13 were dealt ;-p)

so why shouldn't we consider there are 6 outs from 28 cards (52 less 20 dealt + 4 on board) giving 3.7 to 1 odds (instead of 9 outs from 46 giving 4.1 to odds)

[Octopus]

[ QUOTE ]
If I have 2 hears in my hand and there are 2 on the board at 4th street, why is it assumed there are 9 outs available?
If, on average, 1 in 4 cards is a heart, and 20 cards were initially dealt to the 10 players, then on average 5 hearts were dealt and I have 2. This leaves 3 dead in various hands and 2 on the table leaving 6 outs (on average - it is of course possible all 13 were dealt ;-p)

so why shouldn't we consider there are 6 outs from 28 cards (52 less 20 dealt + 4 on board) giving 3.7 to 1 odds (instead of 9 outs from 46 giving 4.1 to odds)

[/ QUOTE ]

Of the 46 unseen cards, each is equally likely to be dealt on the river. Your assumptions about the "average" distribution of the other hearts are no relevant to that fact.

[Delphin]

You only have information about 6 cards for sure at this point. You know your two cards and you know the 4 board cards. You have no information about any other cards. The 9 hearts that are out there are all possibilities on the river. There are 9 possible hearts that might be on top of the deck out of 46 cards that might be the top card in the deck. You have no reason to suspect that any card has any higher probability of being the top card on the deck compared to any other unseen card.

If this is unintuitive to you, there is a mathematical excercise you can do to prove it to your self. Figure the probability that no hearts are in the deck (all are in players hands) and multiply this by the probability that you make your flush on the river (0/28). Then figure the probability that 1 heart is in the deck (rest in players hands) and multiply this by the probability that you make your flush on the river (1/28)...do this for every number of hearts that might be left in the deck until you have calculated the probability that there are 9 hearts left in the deck and multipled by the probability of making your flush on the river (9/28). When you add up all these joint probabilities, you will get the total probability of making your flush on the next card. You will get exactly 9/46, and maybe then you'll believe that every card is equally likely to be contained in the deck, and none of this extra work was necessary. If you have to do it once to believe it, I don't blame you.

You cannot just assume that a certain average number of hearts are left in the deck. The reason you got the different numbers is because your average is wrong. There is a probability associated with each possible # of hearts left in the deck, and when you take a weighted average of that number times the probability, you should get 5.478/28 which is the same as 9/46. The reason that the number is lower than 6 is that you didn't take into account that you already have some evidence that the hearts are not evenly distributed in this hand. You know that you got two and the board got two, therefore there is less than the "average" number left in the deck.

[@bsolute_luck]

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so why shouldn't we consider there are 6 outs from 28 cards (52 less 20 dealt + 4 on board) giving 3.7 to 1 odds (instead of 9 outs from 46 giving 4.1 to odds)

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why even spend the time thinking about this? the difference between 4.1 and 3.7 is well pointless really. i spend my time thinking of better, more important things like how i'm going to spend my winnings when my flush comes in [img]/images/graemlins/laugh.gif[/img]

[wireMan]

Very nice explanation Delphin. Didn't do the math, saving my energy for mowing the lawn in a short bit, but I'm trusting you on this one.

[aK13]

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Very nice explanation Delphin. Didn't do the math, saving my energy for mowing the lawn in a short bit, but I'm trusting you on this one.

[/ QUOTE ]

Probability is a kick in the balls.

I calculated the odds of drawing a full house in 5 card draw...wow that was a bitch...and I don't even remember what it was.

I also calculated some other gambling games...turns out craps is the highest chance to win (assuming you can't count cards at blackjack tables, and don't play poker) at a whopping 49.5% or something.