[BruceZ]

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Let us assume hole-cards + flop give me 5-6-7. What are my chances to complete the straight on the river?

This is what I have tried so far:
4/47 = 8,5%

now I can hit the upper or the lower end of my straight:

8,5% x 2 = 17%

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You can hit both the upper and lower end by the river, and this possibility is getting double counted when you multiply by 2 here, and then multiply by 2 again at the end of the calculation.


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then I have to hit the remaining card:

4/46 = 8,7%

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8/46 = 17.4%.


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then I multiply 17% x 8,7% = 1,48%

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17% x 17.4% = 2.96%


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in the end I double that because I have two ways to hit my cards. Like 9-8 and 8-9 for example.

1.48% x 2 = 2.96%


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2.96% x 2 = 5.92%

Now we must subtract off the probability of 48 or 84 that got double counted.

5.92% - (8/47)x(4/46) = 4.4%.


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The correct number should be 4%, so where is the mistake?

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Here is how I'd do it. The possible 2 card combinations that make a straight are 34, 48, or 89. Each of these has the same probability, and they are mutually exclusive, so the total probability is 3 times the probabilty of any one of these, or:

3*(8/47)*(4/46) = 4.4%.