Good afternoon, I find many topics on how to calculate the probability of a full house, a straight, and so on. But I did not find a source where the counting proceeds with the knowledge of some cards. Suppose you have $2$ and $3$(you can have $5$ cards) of the same suit in your hand, what is the probability that you will have a straight flush?
$$\frac{?}{\binom{50}{3}}$$
I think there are many shortcomings in this formula, please help me understand.
How did I get such a formula according to my logic. We were dealt two cards, now there are $50$ cards left in the deck and 3 possible combinations. The total number means the combination is $\binom{50}{3}$. Next, we need to find the number of straight flushes we can get from $2$ and $3$ of the same suit. If you count on your fingers, it's easy: $A2345$, $23456$, which means only two combinations. But how do you find them using the formula? I don't know, so I wrote a questionmark on top.
You want to count ways to obtain a straight flush that contains the 2 and 3 of the same suit that you know that you have in your hand.
However, you have done that! Don't do more than you have to. If you can easily count things, just do so.
Thus the probability for obtaining one from the two such hands, given that you have a 2 and 3 from the same suit, when selecting another three from the fifty remaining cards, is:
$$\dfrac{\dbinom 21}{\dbinom {50}{3}}$$