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The Rule of 4 and 2 multiplies the out count by four on the flop and by two on the turn. It is a linear approximation, and it always lands high: 9 outs gives 36% against an exact 34.97%, while 15 outs gives 60% against an exact 54.12%. The poker odds chart is where the exact numbers live. What the shortcut is actually computing The rule treats each out as one independent chance on each card still to be dealt. Two cards remain on the flop, so every out is counted across two streets; one card remains on the turn, so it is counted once. That is where the four and the two come from. It is a count of shares, not a probability calculation, and shares add cleanly while probabilities do not. ...

Three of a kind outranks two pair because it is rarer: 54,912 five-card combinations against 123,552. The hand rankings chart places them at ranks 7 and 8, and the probability column is the reason for that order. Rarity sets the order Poker hand rankings are an ordering by how often each hand occurs among all five-card combinations. A hand that appears less often beats a hand that appears more often. The table lists 2.112845% for three of a kind and 4.753902% for two pair, so the rarer hand is the one with the smaller share of all five-card hands. ...