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A "Straight" and a "Flushes" are "Poker Hands". Which is a specific class of selection of $5$ cards from a standard $52$ card deck (no wild cards) the details of which can be found here.

In classic $5$-card stud poker each player must make a poker hand out of $5$ cards which they are dealt. In this game the probability for obtaining a straight and flush as defined above is:

$$P_{s_{5}}=\frac{{10 \choose 1}{4 \choose 1}^5 - {10 \choose1}{4 \choose 1}}{ 52\choose 5} \approx 0.76\%$$

$$P_{f_{5}}=\frac{{13 \choose 5}{4 \choose 1}- {10 \choose1}{4 \choose 1}}{ 52\choose 5} \approx 0.367\%$$

However the game of poker everyone is familiar with (Texas Hold'em) is a $7$-Card Poker Game and so its probabilities are equivalent to $7$-Card Stud wherein each player must make a poker hand out of $7$ cards which they are dealt. In this game the probability for obtaining a straight and flush is:

$$P_{s_{7}}\approx 10.4 \% $$

$$P_{f_{7}}\approx 5.82\%$$

The calculations for 7-Card Stud Probabilities can be found here.

We can also see for $10$-Card Stud the probabilities are:

$$P_{s_{10}}\approx 17.68 \% $$

$$P_{f_{10}}\approx 19.12\%$$

Obtained from here, where you can also find probabilities for all other stud games under 10.

Something particularly interesting I noticed is for more than 9 cards the probability of obtaining a straight is smaller than that of a flush, however in the actual game of 9-Card Stud a Flush is regarded as more valuable than a straight.

I am interested to know is there an explicit formula someone can find for the probability of obtaining a straight/flush in $n$-Card Stud Poker?

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  • $\begingroup$ I do think you could try to derive the formulas by yourself by looking at the formula of 5-stud poker hands... The expressions are similar. $\endgroup$
    – JetfiRex
    Commented Jan 23, 2022 at 5:10
  • $\begingroup$ What is your math background and what is the source of the problem? $\endgroup$ Commented Jan 23, 2022 at 5:33

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