odds of drawing into a pair?

This seems simple enough. But am I simply wrong?:

Say I hold a Jack, and draw four cards. I figure I have four chances of drawing at least one more jack from the remaining cards,or:

3/47 + 3/46 + 1/15 + 3/44 = about 26.4% chance. Did I do that right?

That method gives a very rough approximation, but it is not really correct. I think to find the chances of "at least one more jack" it is easier to subtract from 100% the chance of not getting any more jacks.

P= 1 - (44/47)*(43/46)*(42/45)*(41/44) = 0.2389 = 23.89%


Edit: My original post had an error in the calculation. Now corrected, I think.

Thanks for your help. It looks like I have a lot of studying to do. : )