Some people say mucked cards cant affect the odds of something happening like a flush If you hold 2 diamonds and there are 2 in the flop because every unseen card is considered to still be in the deck. But i mean there is a difference between those cards and the ones in the deck. Mucked cards cant show up on flop turn river right? So if youre in a table of 9 and 7 people fold why not calculate the average number of diamonds in the mucked hands and discount from the total of outs before multiplying by 4? I mean 9/47 (2 in your hand 3 in flop) x 14 folded cards.. so theres an average of 2.68 diamonds in the mucked hands which is -10% likely of a flush being made.

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Actually x16 because there could be diamonds in the villains hand too

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There are even more cards in other people's hands that do not make your flush, but those cards aren't factored in, either, so it all works out.

The only time you should be factoring in mucked cards is if they have been exposed, since you now know for a fact that the odds have been affected.

But why would they be factored in? I mean they dont help me improve. Id be analyzing strictly the best hand possible and its odds. What i mean is people overestimate too much their odds when they need outs to win

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What you’re not factoring in is that you’re changing the numerator of the fraction (how many diamonds there are) but not the denominator (how many total cards there are).

Imagine a simplified scenario with two red cards and two black cards. The odds of drawing a red card is clearly 50%. Now, muck one of house cards at random; on average we are mucking half a red card. Does this change the odds that you then draw a red card? Walk this answer through, and see if you get it.

If you have two diamonds in your hand and there are two on the flop, isn't it more likely that the other players have spades, hearts and clubs instead?
If you know five cards (your holecards and the flop) and four of them are diamonds, there are still 9 more diamonds out there, but you don't know if they are in other players' hands, got folded, or are still in the deck, because the deck was shuffled randomly. There are 47 cards shuffled in a random unknown order. 9 of them are diamonds, so the turn card has 9/47 chance of being a diamond.

Think of it this way when you want to discount mucked cards:
If there are mucked diamonds the chances of a diamond flush decreases...but when there are no mucked diamonds the chance of a diamond flush increases.

Let's do a simple example:

2 player table. You have deuces and are setmining. The other player folds (yeah, I know - you win...but bear with me)
Chances of you hitting a set on the flop if we just use the regular approach (treating mucked cards as part of the deck):
1 - 48/50 * 47/49 * 46/48 = 11.75%

If we consider mucked cards as separate from the deck then we have 3 cases:
i) Folding player mucked 2 deuces
ii) Folding player mucked 1 deuce
iii) Folding player mucked no deuces

So we calculate the chance for each case and the chance that leaves you with hitting a set - and then add up the three cases.

i) Chance that the folding player was also dealt 2 deuces (since you have the other two deuces there is removal): 2/50 * 1/49 = 0.08%
Chances that you hit a set on the flop under these conditions: 0% (no deuces left in the deck)

ii) Chance that the folding player was dealt 1 deuce: 2! * 2/50 * 48/49 = 7.8%
Under these conditions your chance of hitting your set on the flop (48 cards remaining in the deck without the mucked cards and the cards in your hand) is
1 - 47/48 * 46/47 * 45/46 = 6.25%

iii) Chance that the folding player was dealt no deuces: the other 92.12%
Under these conditions your chance of hitting a set (or quads) on the flop are
1 - 46/48 * 45/47 * 44/46 = 12.23%

Adding i) and ii) and iii) up we get
0.0008 * 0 + 0.078 * 0.0625 + 0.9212 * 0.1223 = 0.1175
i.e.: 11.75%

So you see in some cases of mucked cards your chances dip and in some cases they rise slightly. The two effects cancel each other out.

I got kinda lost in the math but i see your point sometimes the odds of flushing are bigger sometimes smaller and at the end they cancel each other. But the point is where do those odds cancel each other out? It certainly cant be on 9 outs since 9 is as good as It gets considering no one received another diamond. So the 9 outs estimate is very optimistic i think

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I think i got confused.. thankyou for awnsering guys It makes sense now.

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case 1) If you discount mucked cards and all 9 outs are still viable (all diamonds still in the deck and none in the muck) then your odds of getting a flush are higher...because now you're drawing from less cards than 50. You're drawing from 50 minus the number of mucked cards.

case 2) If you discount mucked cards and some diamonds are in the muck your odds are lower... (but you're still drawing from less cards than 50. You're drawing from 50 minus the number of mucked cards in this case too)

adding
the probability for case 1 times the probability of a flush under these conditions
plus
the probability for case 2 times the probability of a flush under these conditions
cancels out.

This is the problem with what you are doing original poster. You are adjusting the numerator to remove the number of diamonds expected to be in the mucked cards, but you aren't reducing the denominator by the number of mucked cards. If you are adjusting the numerator to account for the mucked cards, then you have to adjust the denominator to account for them also.





...

Yes, this issue can be vexing to fully understand. Others above have given reasoning and shown the math why it is correct to treat (unknown) folded cards as essentially still "in the deck".

The central tenet is that your subjective view of the probability of future cards to come is based solely upon the information you have about cards you may know something about (like your own cards, board cards already dealt, etc.).

If you have no information on the cards that Villain 1 has folded, then treating them as part of the pack of unknown cards yields the correct (subjective) probability of future cards.

However, if Villain 1 has folded and you believe that he/she had not folded two diamonds, then this affects your subjective probability of future cards. For example, if you know he/she folded two spades, that changes the percentage of diamonds remaining in the virtual stub from which future cards emanate since you now know that there are two fewer non-diamonds in the virtual stub than in the absence of such information.

Bottom line: your view of the distribution of possible cards to come depends solely upon your view (information, inference, etc.) of the cards that have already been dealt. And if you know absolutely nothing about a card that has already been dealt, it is mathematically correct to treat it as still belonging to the virtual stub.

interesting discussion,

if you have AK or AQ suited with hearts. and 2 people see your pre-flop raise, then yes, i would say they are less likely to have hearts themselves. and alot of lesser hearts will have been folded.......... but i'm trying to work it out in my mind. i think maybe where it matters if your implied odds won't be as good as less likely someone is playing 2 hearts themselves or even one very high heart as you have 2 of the 3 top hearts yourself.......

ultimately i don't think it matters that much as other hearts are as likely to have been folded as still in the deck. as i have implied, a bunch of callers will suggest that there may be marginal more hearts left in deck than simplest card math would suggest.

i had never really thought of your flush angle before....... certainly when everyone has folded to you on button and you raise marginal hand, i think there is moderately higher chance of one of the blinds having very good hand as lots of garbage cards have been folded already. but that's high cards, not suited cards.

Question then is: does the marginally different chance that you have a dominating flush against someone after you affect how (or even whether) you play the hand preflop? not really, so I wouldn't worry.

Yes, card removal of a specific suit can be crucial. The most amazing play along these lines I've seen was by isildur1 (Victor Blom) at the final hand of the Partypoker Millions in 2018. Here's the hand:
https://www.youtube.com/watch?v=ZCyv3RQ7kuo
...where it was absolutely crucial that the king he held was the king of hearts (any other king he would almost certainly have folded)

Yes .. There is the math ... and then there's the hand being played, which includes the Board and player 'information' received during play. There are plenty of little influences to consider ...

Which 2 are in your holding? Which 2 are on the Board?

These Boards are WAY different ... You hold 9T
AJ4 v A67
Or
A54 v A83

Against which Boards is your flush draw more likely to be good?
How do you view betting players and/or calling players during these hands?
How do you view players who double check their cards and sigh muck when facing a pot-sized bet?
Did the PF raiser fold out and you are now facing a limp/caller?
How do you view these Boards when holding a naked A or K of ?

These are factors that may allow you adjust 'available' outs or opponents holdings .. not 'well some might be in the muck'.

Have you heard of this game called PLO? There are more cards in the Muck than the Stub once the Flop comes out .. .can't let that affect your 'true math'. GL

Consider a full deck of cards with no cards dealt yet. Everyone probably agrees that each card in the deck has a 1/4 chance of being a diamond.

The deck is one big pile, but now make piles of two. If you keep all the cards face-down, how will the mere act of making piles of two change any card's probability of being a diamond?