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Given that you are dealt the J♥ and Q♥, what is the probability that you will eventually make a straight flush or royal flush?

Having these cards in your hand means that:

1). the only way you could possibly make a straight flush is if your hand becomes:

  • 8♥ 9♥ T♥ J♥ Q♥
  • 9♥ T♥ J♥ Q♥ K♥

2). the only way you could possibly make a royal flush is if your hand becomes:

  • T♥ J♥ Q♥ K♥ A♥

I'm having a difficult time in figuring the probability of this.

5

There are five board cards in hold'em. Since you start with two known cards, there are 50 unknown. That means there are 50x49x48x47x46 ways the board can come. Since the order of the cards on the board doesn't matter, divide that by the number of ways 5 cards can be arranged (120), that's 2118760 total distinct boards. There are 47x46/2 of those boards that contain the three cards we need, so dividing again we get 1960. So 1 out of 1960 times we'll eventually make the royal. There are 46x45/2 boards that make each straight flush (with no ace--else they'd make the royal as well), so that makes 1 out of 1023 5/9 for the straight flushes.

  • This answer is very close, but the 47x46/2 only applies for the royal flush. For the straight flushes you should use 46*45/2, because you can't have the ace on the board for the king high SF and you can't have the king on the board for the queen high SF. – Paul Mar 2 '15 at 17:03
  • Oops, you're absolutely right. – Lee Daniel Crocker Mar 2 '15 at 17:09
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More complex then you would think.

Board has no other hearts

+3 x combin(39; 2) / combin(50; 5) = 3 x 741 / 2118760 = 0.00105 = 1 / 953

But the board could also be a 4 hearts 8♥ 9♥ T♥ K♥ You cannot just change the the 39 to 50 as it is double dipping. So the odds are very slightly better than the number above.

3 more ways to make

               j    q       
    8   9   T           k   
        9   T           k   a
7   8   9   T               

3 x 39 adds 117 for 0.0011 = 905.453

And there are 3 5 heart boards = 0.0011 = 1 / 904.29

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