Odds of a 3 card straight

You made no assumption on hero's holding, so we shall stay with that.

From a probability perspective flopping a straight is the same case as having a straight in 5-card stud; that is, five dealt cards make a straight.

There are 10 different straights (high card Ace down to high card Five) and any suits will work for each of the five cards, except we want to exclude flushes (straight flushes).

Thus, the number of straight combos is: 10*(4^5 - 4) = = 10,200

Total number of 5-card hands is: C(52,5) = 2,598,960

So prob of flopping a straight is: = 10,200 / 2,598,960

= 0.392%

Of course, if you want to assume a holding, the probability will be different. For example, if you have A 8, you cannot flop a straight but holding 76s has probability = 6*(4^3-1)/C(50,3) = 1.928%.

A Google search will result in several sites that solve this problem for a number of different assumptions on hero's holding.

Sorry I wasn’t clear. I meant the flop having a 3 card straight. Hero has no cards in this hypothetical.

My answer is different from the result I found which is why I posted my result.

FWIW, using an exhaustive sum approach, I get 3.4751%, same as you.

Your result is correct if you include a 3 card straight flush, e.g., 4h 5h 6h.

If straight flushes are to be eliminated, the total number of successful combos is 12*(64-4) = 720 so the hit probability is 3.258%

That explains the discrepancy. Thanks guys.