Calculating the odds of drawing two pairs OR three of a kind when drawing five cards from a deck

 : A (52:52*)

A: A (3:51*), B (48:51*)
     (1:17)    (16:17)

AA: A (2:50 = WIN), B (48:50*)
AAB: A (2:49 = WIN), B (3:49 = WIN), C (42:49*)
AABC: A (2:48 = WIN), B (3:48 = WIN), C (3:48 = WIN), D (40:48 = LOSE)

AB: A (3:50*), B (3:50*), C (44:50*)
ABA: A (2:49 = WIN), B (3:49 = WIN), C (44:49*)
ABB: A (2:49 = WIN), B (2:49 = WIN), C (44:49*)
ABC: A (3:49*), B (3:49*), C (3:49*), D (40:49 = LOSE)

ABAC: A (2:48 = WIN), B (3:48 = WIN), C (3:48 = WIN), D (40:48 = LOSE)
ABBC: A (3:48 = WIN), B (2:48 = WIN), C (3:48 = WIN), D (40:48 = LOSE)
ABCA: A (2:48 = WIN), B (3:48 = WIN), C (3:48 = WIN), D (40:48 = LOSE)
ABCB: A (3:48 = WIN), B (2:48 = WIN), C (3:48 = WIN), D (40:48 = LOSE)
ABCC: A (3:48 = WIN), B (3:48 = WIN), C (2:48 = WIN), D (40:48 = LOSE)


AAA: 1:17 * 2:50 = 1:425 = 4704:1999200
AABA: 1:17 * 48:50 * 2:49 = 96:41650 = 48:20825 = 4608:1999200
AABB: 1:17 * 48:50 * 3:49 = 144:41650 = 72:20825 = 3456:1999200
AABCA: 1:17 * 48:50 * 42:49 * 2:48 = 4032:1999200 = 6:2975
AABCB: 1:17 * 48:50 * 42:49 * 3:48 = 6048:1999200 = 9:2975
AABCC: 1:17 * 48:50 * 42:49 * 3:48 = 6048:1999200 = 9:2975

ABAA: 16:17 * 3:50 * 2:49 = 96:41650 = 4608:1999200
ABAB: 16:17 * 3:50 * 3:49 = 144:41650 = 6912:1999200
ABBA: 16:17 * 3:50 * 3:49 = 144:41650 = 6912:1999200
ABBB: 16:17 * 3:50 * 2:49 = 96:41650 = 4608:1999200

ABACA: 16:17 * 3:50 * 44:49 * 2:48 = 4224:1999200
ABACB: 16:17 * 3:50 * 44:49 * 3:48 = 6336:1999200
ABACC: 16:17 * 3:50 * 44:49 * 3:48 = 6336:1999200
ABACA: 16:17 * 3:50 * 44:49 * 3:48 = 6336:1999200
ABACB: 16:17 * 3:50 * 44:49 * 2:48 = 4224:1999200
ABACC: 16:17 * 3:50 * 44:49 * 3:48 = 6336:1999200

ABCAA: 16:17 * 44:50 * 3:49 * 2:48 = 4224:1999200
ABCAB: 16:17 * 44:50 * 3:49 * 3:48 = 6336:1999200
ABCAC: 16:17 * 44:50 * 3:49 * 3:48 = 6336:1999200

ABCBA: 16:17 * 44:50 * 3:49 * 3:48 = 6336:1999200
ABCBB: 16:17 * 44:50 * 3:49 * 2:48 = 4224:1999200
ABCBC: 16:17 * 44:50 * 3:49 * 3:48 = 6336:1999200

ABCCA: 16:17 * 44:50 * 3:49 * 3:48 = 6336:1999200
ABCCB: 16:17 * 44:50 * 3:49 * 3:48 = 6336:1999200
ABCCC: 16:17 * 44:50 * 3:49 * 2:48 = 4224:1999200

Total sum: 136416:1999200 = 6.82%




How about the odds of getting a flush if you keep three cards of a suit:
10:47 * 9:46 = 90*2162 = 4.16%


Conclusion:
In five-card draw you've got about a 13% chance of getting two pair or three of a kind between your two hands, if you don't keep anything.
Keeping a random card will increase your chances slightly.  Keeping a pair will increase your chances significantly.

If you do not have a pair and do have three of the same suit, it's likely not worth it to go for the flush.
 - UNACCOUNTED FOR:  On the second draw, five unique cards are missing, potentially making it harder to get a pair/3ofK