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Your calculations are off. You have to remember that you can't just add probabilities when they are dependant on each other. It might work well for a quick approximation of a simple probability, but when the calculations get more complicated you will be way off.
The chance the opponent has an overcard is NOT 12/50 + 12/49 = 48,5%.
It is: 1 - 38/50 * 37/49 = 42,6%
The chance the opponent has a specific overcard is NOT 4/50 + 4/49 = 16,2%
It is: 1 - 46/50 * 45/49 = 15,5%
The chance the opponent overcard will hit is NOT (assuming he has EXACTLY one specific overcard) 11/48 + 11/47 + 11/46 = 70,2%
It is: 1-45/48*44/47*43/46 = 18,0%
These numbers can't just be multiplied, since you don't take into account if he has 2 of the same overcards.
I'd do it like this:
The chance the opponent has 2 of the same overcard:
(a) 12/50 * 3/49 = 1,47%
The chance the opponent has 2 different overcards:
(b) 12/50 * 8/49 = 39,18%
The chance the opponent has at least 1 overcard:
(c) 1 - 38/50 * 37/49 = 42,61%
The chance the opponent has precisely 1 overcard:
(d) c - b - a = 37,22%
Now you have to look at the chance of one of his overcards flopping if he has 2 identical overcards, 2 different overcards or 1 overcard.
If he holds 2 identical overcards, the chance of one flopping is:
(A) 1 - 46/48 * 45/47 * 44/46 = 12,23%
If he holds 2 different overcards, the chance of one flopping is:
(B) 1 - 42/48 * 41/47 * 40/46 = 33,63%
If he holds exactly 1 overcard, the chance of one flopping is:
(D) 1 - 45/48 * 44/47 * 43/46 = 17,96%
Therefore the chance of the opponent holding an overcard which flops is:
(a) * (A) + (b) * (B) + (d) * (D) = 8,18%
Since you have already lost if he holds 2 identical overcards, it doesn't matter if the overcard flops or not, so the more interesting number is:
(a) + (b) * (B) + (d) * (D) = 9,47%
There might be a simpler way to calculate this, but I couldn't figure it out.
Feel free to correct any miscalculations.
/Rune
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