I've revised the question after further exploration. Here is the information the is needed The probability of a Straight Flush occurring is 0.00001540898529 The probability of a Royal Flush occurring is 0.000001540898529 I have changed the value of the Probability of a Straight Flush occurring overall because there's a rule in Poker that prohibits the use of cyclical hands. Basically, a player can't use the following hands: Jack, Queen, King, Ace, 2 Queen, King, Ace, 2, 3 King, Ace, 2, 3, 4 For comparison, the original probability of a Straight Flush occurring was 0.00001539077 and the probability of a Royal Flush occurring was 0.000001539077. Moving on, I am unsure whether I should be calculating the probability of a Royal Flush occurring FROM a Straight Flush OR the probability of a Royal Flush occurring AFTER a Straight Flush But only taking into account that the hand was removed from a standard deck of 52 cards. I have provided a Tree Diagram to illustrate both situations Please note: If a Royal Flush occurs, I am only taking into account the fact that the first hand dealt was a Straight Flush but not a Royal Flush. However, I am still including the fact that a Royal Flush can occur within a Straight Flush. Please see the first image if you do and don't understand what I mean. Lastly, I am using the equation for Conditional Probability to solve this, which would be P(A|B)=P(A intersection B)/P(B) For those of you that may not remember, P(A intersection B) is just the values of A and B multiplied together Let P(B) indicate the probability of a Royal Flush occurring OVERALL Let P(A) indicate the probability of a Royal Flush given a Straight Flush. Here are the tree diagrams: The probability of a Royal Flush Occurring from the Straight Flush
Also, would I also need to revise my secondary values? Since the first 5-card removal is not replaced?
I hope this clarifies what I need.
Your question, going by the clear header rather than the rambling in the body, is that someone has been dealt a straight flush(excluding royal flush), and given that, you want to know your probablity of getting a royal flush.
Out of $9$ possible such straight flushes from a suit, $4/9\; (23456,34567,45678,56789$) allow you to get a royal flush from that suit, whereas $5/9$ don't
So $4/9$ of the time, chances are open for $4$ royal flushes out of $47$ cards, wheras $5/9$ of the time they are open only for $3$ royal flushes.
Thus P(get Royal flush | straight flush has been dealt)
$$= {\frac49 *4 +\frac59*3\over \binom{47}5}, \approx 0.0002245 \,\%$$