What is the odd of filling 4 suited cards by river to make a flush? This would be starting from the "flop" in hold 'em, and starting from the fourth card, all of which are suited, in seven card stud.
Your question can be re-written like this:
What are the odds of having 4 cards with the same suit, from the 5 available community cards?
The answer can be found by using basic probabilities and computing probabilities of composed events.
You can have 5 boards in which 4 cards are of the same suit:
Here, X means the suit you're asking for, Y means any other suit. Of course, each of the 5 scenarios have the same likelihood of occurrence.
There are 13 cards of a suit. We'll take out the two preflop cards, which means that we have 12 cards out of 50 in the deck (suppose you're drawing for a flush, otherwise you wouldn't have asked this question). So, one of the 5 scenarios described above occurs with the probability of:
(12/50) * (11/49) * (10/48) * (9/47) * (38/46) =0.00177556684098246143971001906776
Because we have 5 scenarios, each having the same probability of occurrence and each one being independent of the other 4 ones, the answer you are seeking for is:
So, you will have exactly 4 cards of a given suit, on the river, in 0.88% of the cases.
If you consider all community cards coming up hearts, this is 50*49*48*47*46/12*11*10*9*8, which is about 0.03%.
The odds of hitting a five flush, given four suited cards on the "flop" in hold'em are about 35%. They are 47% starting from four cards, all to a flush, in seven card stud.
Your chances of eventually making the flush go down with each new card that does not "hit" the flush. If it "hits," you chances go up to 100%.
Actually four to a flush (or straight) is GOOD "nothing." It's actually rarer (and potentially more valuable) than a pair, and is "nothing" just by the rules of the game.
The first answer above is wrong as it ignores the odds of the fifth card not being of the same suit.
To get the odds of getting four cards of a particular suit and one of another suit, use the hypergeometric distribution with 4, 5, 12 and 50 as parameters. The answer is about 0.88%. We assumed your two cards are off-suit so you can double this to get about 1.76%.
Note this doesn't include the possibility of all five community cards coming up the same (specific) suit. This is about 0.0374%
If I understood your question correctly, you have 4 cards of the same suit at the flop and you want to know what are the odds of hitting the flush by the river.
first just random flush
so now to the question - you do not hold suited cards - get 4 or 5 to match