Making best hand on the river

[PrayingMantis]

This is from a discussion I had recently on the psychology forum, under the thread "The river card syndrome".

There's a question I still feel curious about, although I was given an answer. I would like to hear the opinions here.

I'll put it as simple as possible:

It's holdem, and you're dealt 2 hole cards. Then the flop is dealt and the turn. It doesn't matter how many opponents play against you, so let's say you're alone.

What is the probability of the river improving your hand (i.e., making your best hand)? And how do you calculate it?

Examples, although I guess you completely understand this:

1. you hold 79, board: 79TJ, river: Q. The river improves your hand. river: 9, it improves. river 5, doesn't improve. etc, etc.

2. You hold QQ, board Q79K, river: 8. The river does not improve your hand. river A improves. Q improves. 2 doesn't. etc.

Flushes and straights count, everything counts.

Thanks,

PrayingMantis

[Henke]

It's quite simple. Let's say x cards improve your hand. There are 46 unseen cards. The probability that one of the x cards come on the river is x/46.

[PrayingMantis]

Thanks for the answer, but this is completely *not* what I'm looking for.

I'm looking for the general probability. Not the calculation of the specific outs for a specific hand, which is a very obvious and basic prodecure.

I'll try to phrase it differently:

if you are all-in, no matter with what cards, what is the probability that you'll make your "best hand" on the river? (as opposed to making it already on the flop, or on the turn?).

The answer should be a number between 0 and 1. .

I hope now it's more clear. You might want to look through the original thread in the psychology forum ("the river card syndrome"). This very question is discussed between PDosterM and myself.

PrayingMantis

[PrayingMantis]

I'm afraid the original question is maybe not well phrased. I will rephrase it:

You are all-in, no matter with what cards. What is the probability that you'll make your "best hand" on the river? As opposed to making it already on the flop, or on the turn?

(Important: By "best hand" I mean *your* best hand, out of all the possibilities your hole-cards and the board give you. I don't mean "best hand" as of the hand that wins the pot.)

The answer should be a number between 0 and 1, not an equation with x to signify outs.

I hope this is more clear.

Thanks,

PrayingMantis

[BruceZ]

This is the same as asking for the probability that the river card will play as part of your hand. If you assume that you are going all-in with any two random cards pre-flop, the answer is 5/7. Before the deal, each of the 7 cards has an equal 5/7 chance of being part of the best 5 card hand.

In reality, you often make the decision to go all-in based on your starting 2 cards. In this case, there is a greater chance that your starting 2 cards will play, and consequently a lesser than 5/7 chance that the river card will play, and the answer will depend on the cards. For example, it is more likely that your hole cards will play if they are AA than if they are 23. If we assume that both of your hole cards play, then the river has a 3/5 chance of playing. I noticed that you got 3/5 as part of your 200 hand "simulation". I suspect that you did not properly read these hands in the cases where one or both of your hole cards did not play.

[PrayingMantis]

Thanks for the answer.

Yes, 5/7 is the answer that was given to me on the other thread.

I don't think I've made such a significant mistake in reading my 200 hands "simulation". But I can't argue with the possibility I did miss a few hands. Although it looks quite strange, that I made a consistent one-side mistake. Well, It was just a little test, to feel the spread of it.

So now a technical question: if I want to check all sorts of things with a cards-deck, is there any good simulator? on-line maybe? I don't mean twodimes, I mean something like a dealer, that can generate cards as in a game, so I don't have to do it manually.

Do you know of any?

Thanks,

PrayingMantis