Flush chances (math)

Total cases: 50C5 = 2118760
There are 52 cards in the deck and you already have 2 of them in your hand. So, there are 50 cards left to be dealt. Flop+ turn + river equals to 5 cards, so you have a total of 50C5 combinations possible.

Favorable cases: 11C3 x 39C2 + 11C4 x 39C1 + 11C5= 135597
There 13 cards of each suit available. Once you have 2, 11 cards of the same suit remain. From the 50 cards available, 39 of them are from the other suits.
There are 3 ways of making a flush postflop:

1º - You need 3 more cards of the same suit to complete your flush.
Flop+ turn + river equals to 5 cards, so there's 11C3 x 39C2 combinations possible (you choose 3 of your suit and 2 of any other).

2º - You can also make a flush with 4 cards of your suit on the table and 1 from any other suit (11C4* 39C1).

3º - Finally, all the 5 cards can be from the same suit (11C5*39C0 = 11C5).

Probability: Favorable/ total = 135597/2118760 = 6,4% chance.

Note: I don't know if you are familiarized with probability concepts, but do a little research and you'll find out what combinations and permutations mean and how to apply them.
Another interesting things to know about flushes, assuming you have 2 hole cards of the same suit:

- Completing a backdoor flushdraw you catch OTF by the river: (13-2-1) / (52-2-3) x (13-2-1-1) / (52-2-3-1) = 10/47 * 9/ 46 = 4,2% chance

- Completing a flushdraw you catch OTF by the turn: 9/47 = 19,1% chance
You have 9 outs and there are 47 cards on the deck to be dealt for the turn.

- Completing a flushdraw you catch OTF by the river: 9/47 + (52-13-1)/47 x 9/46 = 34,97%
You can hit OTT (9/47) OR (+ sign) you can miss OTT (52 cards minus the 13 cards from the suit (2 of yours and 11 remaining) and the card that was dealth on the flop board from another suit) and hit OTR (9 outs with 46 cards possible)

Completing a flushdraw you catch OTT by the river: 9/46 = 19,6% chance

Being dealt 2 cards of the same suit: 12/51= 23,5% chance
You are dealt a random card and all we need to know is the probability of the second card being the same suit as the first.
12 cards of the same suit remain, of 51 total cards in the deck.

Hi LuNaNDu,

I did some googling and found out that it is 6.5% probability to hit a flush from pre (with 2 same suit) to river.

Originally posted by zetozinho
Being dealt 2 cards of the same suit: 13C2 / 52C2 = 5,9% chance

Times four! => 23.5
4C1 * 13C2 / 52C2

But else great job!

(with maybe two very small caveats)

The question posted did not include if you have two of the same suit on the flop or not? Is it implied the flop has two of the same suit as your two suited holdings?

Otherwise how can a person hit a flush if there is not at least one of the same suit is on the flop?

What would be a situation of needing to hit runner or runner or not?

I look at the fallacy of making flushes.

This is why I keep it simple, 5-1 to enter to then acquire the right amount of bets when making a flush, with the right amount of players to then get one or more to pay off the right amount of bets.

9 outs 47 unseen cards on the flop.

If not hit on the turn, then 9 outs 46 unseen cards.

5-1, this is my maxim to enter hands or stay in hands unless the dynamics allows my game to adjust.

The right % to hit flushes is 20%, and, I learned this in college about fallacy of gamblers

This is why I bet my draws in the right situations, stay in hands with the right odds and predictable players in others.

Others may disagree, but this is why it is fun to discuss because each of us have to determine what is of value depending dynamics of players in certain situations .

Good luck.

Originally posted by Rhodriguez

Originally posted by zetozinho
Being dealt 2 cards of the same suit: 13C2 / 52C2 = 5,9% chance

Times four! => 23.5
4C1 * 13C2 / 52C2

But else great job!

(with maybe two very small caveats)

I came to the post to correct that! Thanks.

Originally posted by STLfan
The question posted did not include if you have two of the same suit on the flop or not? Is it implied the flop has two of the same suit as your two suited holdings?

Otherwise how can a person hit a flush if there is not at least one of the same suit is on the flop?

I think that information was left out on purpose. We can make a flush if there is just 1 same suit card OTF. The questions was, what is our probability to hit a flush by the river if we hold 2 same suits preflop.

Dear la55i,

I understand and was just stating the obvious, however, if one does not go over in ones mind what the person is posting or observing at the table for example, the reason why one would chase a 4% chance to hit runner runner or when pre it is 8-1 to hit a four flush draw, thus, giving 9 outs, then it is more than just stating the obvious it becomes clear of how the odds miss-lead a player to how the odds and percentages are not followed because of the post itself.

To play within the odds because of the math is just the first part of understanding of why would one be involved past the odds and percentages, but to recognize when a player is not getting the right odds and percentages based on how many bets it takes to make a call pre to staying in the hand until the river.

This is why I stated the obvious, not only to ask the question but to further the conversation of why would one chase a flush past the basic math of making a flush, do players really think of why they are in the hand but just merely chase based on the math?

Do they understand the dynamics of the player(s) because the fallacy becomes a vestigial habit, meaning a player will assume they will get paid off, but, in low limit games a person can tip off their hand by the way they play the hand. Does this not go hand in hand from the original thought?

Without enough players in a hand the knowing if someone will get paid off 10% or 20% of the time by player X, the brain does not know the difference. Getting paid off by the right opponent(s) becomes more important than just the first part that math that is empirical of making or chasing a flush or any hand for that matter.

So, it becomes more than just the post itself, and this was my point. Thank you for questioning my thoughts and I appreciate the chance to converse with you as well and glad to have had the chance to talk to a moderator.

Ah okay, I see that you wanted to have a more in-depth discussion about this I didn't mean to question your thoughts. Just tried to say that maybe this was a simple math questions out of curiosity. But I think you had some good points there