I am learning poker math. But run into an issue that I don't understand

I want to calculate the probability for drawing a set or higher with a pocket pair (let's say pocket kings - KK) on the flop.

I can calculate this using two methods:

Method 1: calculate the probability of not drawing a set or higher
I know there are two Kings left in the pack: K and K
for the first card there are 50 unseen cards, of which 48 won't give me a set
for the second card there are 49 unseen cards, of which 47 won't give me a set (or higher)
for the third flop card there are 48 unseen cards, of which 46 won't give me a set (or higher)

so calculating this probability = 48/50 * 47/49 * 46/48 = 88.24% of not drawing a set (or higher). I multiply probabilities for each of the three cards here, cause I want the probability of the set (or higher) not being completed by any of the three cards (not the 1st, not the 2nd nor the 3rd card).

Inversing this 1-88.24% results in probability of drawing a set or higher of 11.76%

Method 2: calculate the probability of drawing a set or higher
I know there are two Kings left in the pack: K and K
for the first card there are 50 unseen cards, of which 2 can give me a set
for the second card there are 49 unseen cards, of which 2 can give me a set (or higher)
for the third flop card there are 48 unseen cards, of which 2 can give me a set (or higher)

So calculating this probability = 2/50 + 2/49 + 2/48 = 12.25%
I am adding the probabilities, since each of the individual cards of the flop can bring me the set.

My questions

• What is wrong in my assumptions cause the two are not equal?
• What is the correct method and why?
• Am I correct in my reasoning for multiplying in method 1 and adding in method 2?
• Any other comments?

Thanks,

Geoffrey
student of the math of poker

Method 1 is ok. Method 2 is wrong for a number of reasons. Suppose there are 20 cards that give success. By adding you’ll get a probability greater than 1.0.

If you want to get more into poker math, you should learn to use combinations. A method 2 would be

Pr( set or quads) =[C(2,1)*C(48,2)+C(2,2)*C(48,1)]/C(50,3) =11.76

The numerator first term is the number of flops with 1 king and 2 non-kings and the second term is the number of flops with 2 kings. These values are summed ( OK for mutually exclusive events) and then divided by the total number of flops.