Pocket pair improving to three of a kind on flop - 12%, 1 in 8, 7:1 No pair hand preflop improving to a pair on the flop (either card) - 32%, 1 in 3.125, ~2:1. If you have suited cards, two will flop - 11%, 1 in 9, 8:1 One pair on flop improving to two pair or three of a kind by river - 22%, 1 in 4.7, ~4:1 Pocket pair improving to three of a kind after flop - 9%, 1 in 11, 10:1 Two over cards improving to a pair by river - 26%, 1 in 3.9, ~3:1 Two over cards and a gutshot improving to pair or straight - 43%, 1 in 2.3, 4:3 Gutshot straight draw hitting by river - 17%, 1 in 6, 5:1 Gutshot and pair improving to two pair or better - 39%, 1 in 2.6, 3:2 Backdoor Flush hitting (5s6s Flop 7sAh9c KsJs) - 4%, 1 in 24, 23:1 Runner Runner Straight (56 Flop 3TQ 47) - 1.5%, 1 in 68, 67:1 Backdoor Flush and One Over Card improving to that pair or flush - 17%, 1 in 6, 5:1 Catching Ace on turn or river (A4 Flop Q63 KA) - 13%, 1 in 8, 7:1 Backdoor Flush and Gutshot improving to one by river (Ac4c Flop 3s5cKs) - 21%, 1 in 4.8, 3.8:1 Backdoor Flush And Two Over Cards improving to pair or flush - 30%, 1 in 3.3, 2.3:1 5 players on flop, that someone has an A when one is on board 58%, 1 in 1.7 4 players on flop, that someone has an A when one is on board 47%, 1 in 2.1 3 players on flop, that someone has an A when one is on board 35%, 1 in 2.9 2 players on flop, that someone has an A when one is on board 23%, 1 in 4.3 3 of one suit on board and another coming (QsTs2s) if you have one, 39%, 1 in 2.6, 3:2 5 players in with board paired, chance of one of them having it 43%, 1 in 2.4 4 players in with board paired, chance of one of them having it 34%, 1 in 3 3 players in with board paired, chance of one of them having it 26%, 1 in 4 2 players in with board paired, chance of one of them having it 17%, 1 in 5.8
Random thoughts from a lawyer, an accountant, a commodities trader, an ex-Marine and a WSOP Main Event money finisher that don't know as much as they wish they did...