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In draw poker with five cards, if I have four cards with an open-ended straight draw, why is it a lower probability to get the fifth card for the straight than if you have four cards in a flush and you just need one more? A straight is not worth as much as a flush, so why is it easier to reach once you have four cards? What is the cross-over point in number of cards, that is, does having three cards to a straight or flush make a straight easier to reach, or is it only when you have two cards that a straight is easier to reach?

This phenomena seems counter intuitive since given five random cards, a flush is harder to get, and thus a more valuable hand than a straight, can someone explain what is going on that causes this cross-over intuitively other than just giving the math probabilities that logically proves it?

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For an open-ended straight draw, there are exactly 8 cards that will give you your straight: 4 of each suit at the low end, and 4 of each suit at the high end. However, to make a flush, you have 9 cards available that will make your flush, as there are 13 cards to a suit, and you have 4 of them in your hand.

At 3 cards to the set (again assuming an outside straight), you now have 16 cards that will help you make your straight (four of each suit for the two cards over, and four of each suit for the two cards lower, assuming you don't have a king or 3 as part of your three). Not every combination of those 16 will be applicable (if you have 5-6-7, a 9 and a 4 won't help, for example), yet with a flush, there are only 10 that will help you (13 to the suit minus the 3 you already have).

All in all, at 3 to a straight there are 48 combinations of cards that will make your straight (assuming 2 possible over cards and 2 possible under cards). At 3 to a flush, though, there are only 45 combinations of cards that will make your flush.

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@WilliamKF I added in the possible card combinations that will make the straight and flush hands to clarify. –  Beofett Jan 10 '12 at 20:47
    
So if you have 3 cards to straight with a gap, e.g. 3,4,6, it's still harder to make your straight than to make your flush with 3 suited cards because there are only 32 combinations that will make your straight. (Is that right math?) –  jhericks Jan 17 '12 at 19:28
    
@jhericks as I understand it, yes. 16 combinations of 2x/5x, and 16 combinations of 5x/7x for 32 outs on the straight. –  Beofett Jan 17 '12 at 19:32
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People are puzzled by this because it is easier to make a straight starting to with two straight cards (8% of the time) than a flush starting with two flush cards (6% of the time).

But GIVEN four of a flush on the flop and four to a straight (open-ended on both sides) it is easier to make the flush (35% of the time from the flop) than the straight (32% of the time). That's because nine cards in the deck will make the flush, and only eight will make the straight.

All this means that the two flush cards on the flop are a greater help than the two cards to the straight on the flop.

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Question refer to 5 card draw poker, there is not flop in this game. –  Tomáš Šíma May 10 '12 at 6:14
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