Thomas R. answered • 05/16/18

Over 25 years of experience and a sense of humor about math

Actually, the answer is slightly uglier than it initially seems, so I understand your worry, Richard.

The good news is that since the smallest card you can draw and still make

**more**than 17 with the 2nd is 8, which means we need not worry about Ace through 7. Combinations that make 17 are:8 with: 10, J, Q, K

9 with: 9, 10, J, Q, K

10 with: 10, J, Q, K (because we already paired it with 8 or 9)

J with: J, Q, K (again, it has already paired with smaller number)

Q with: Q, K (see above)

K with K

For the first row (the row for 8), we have a probability of

**(4/52) * (16/51) =**

For the "9" row:

**(4/52) * (19/51)**For the "10" row:

**(4/52) * (15/51)**For the Jack row:

**(4/52) * (11/521**For the queen row:

**(4/52) * (7/51)**For the king row:

**(4/52) * (3/51)**If you add up all of the probabilities, you can simplify the procedure by multiplying the

**(4/52) = 1/13**by the sum of all of those second factors, which totaled:**71 / 51**In the end, you get

**71 / 663**.Please note that on most lines, we had to lower the probability whenever we picked out a 2nd card of the same kind.