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When you should save the pair and discard the other 3cards

If you discard all but the fours, the total number of remaining cards that could possibly be dealt to you is 52 - 5 = 47. Of those 47, two are fours. There are several possible outcomes that will result in a better hand than the one you currently have:

  1. Getting one four and two unmatched cards (3 of a kind)
  2. Getting two fours and another card (4 of a kind)
  3. Getting a pair of something else and an unmatched card (2 pair)
  4. Getting a pair of something else and another four (full house)
  5. Getting three of a kind of something else (full house)
Each of these events is disjoint from the others, so the probability of improving your hand is the sum of the five probabilities from each item on the list. To compute these probabilities, you must keep in mind that the deck of remaining cards contains only two each of 4; three each of 5, 9, and Q; and four each of the other denominations.

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