Counting the outs when you have a split pair versus an opponents higher split pair

A split pair means when you make a pair with one card in your hand and one card on the board.

For example:

Your hand: ATo Flop: J-T-3 rainbow

In this case, you have a split pair of T's, one in your hand and the other on the board. If you suspect your opponent has a split pair of J's, then you may think you have 5 outs (three A's and two T's). However, if your opponent's kicker matches your kicker, then you would only have 2 outs (two T's). For example, if he had JT, he would already have two pair, and you would have 3 outs (three A's). But if he had AJ with only one pair, and you hit your A for two pair, then he would simultaneously hit his two pair as well, and his two pair would be better than your two pair since he would have A's and J's while you have A's and T's. In this case, you would only have the two remaining T's as outs. This may seem counterintuitive at first because in this situation, you would actually prefer he had JT for two pair than AJ for one pair. You prefer your opponent to have what would appear to be a stronger hand because you would actually have a better chance of drawing out against his two pair of J's and T's than you do of drawing out against his one pair with the same kicker as yours. When you improve by hitting an A, you are counterfeited because your improvement is fake.

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