Odds of making the full house?

Question

Game is Texas Holdem. This scenario actually happened at a mini tournament I was in while camping the other weekend.

I was heads up against an opponent who had about $350 in chips to my $250. I was small blind at $20 and he was big blind at $40.

I had pocket threes, he had 4 5 off suit.

Preflop: I raised it to $70 and he called. Pot at $140.
Flop was 2 3 6 rainbow.

I bet $50, he raised to $100, I went all in, he called.

I see I am a set of threes against 2 through 6 straight.

I wound up getting the full house and winning the tourney (for which I made $25 woo hoo!). What were my odds?

Please give me straight forward logic and don't go to a poker calculator site. I already calculated it myself.

Answer 1

The easiest way to calculate this one is to figure out how many times you will miss.

Even though we didn't know your opponents cards at the time, I will figure them into the equation.

On the flop a 2, 3, or 6 would have improved your hand to the best hand. This means 7 cards would have helped you. There 52 cards in the deck, we know what 7 of those cards are "33, 45, 236". These 7 cards subtracted from 52 equals 45 other possible cards to come on the turn. We subtract the 7 cards that will improve your hand to the best hand and we come up with 38 cards that do not improve your hand to the best. After the turn you now have a 2, 3, 6 or another of the same rank of the turn card to improve your hand to the best. So after the turn you have ten outs that will win the hand for you. There will be 44 other possible cards that could come on the river minus ten cards that give you a full house or quads. 34 cards do not help you on the river. We miss the turn 38/45 times and miss the river 34/44 times. Multiply these together and you know how often you won't get a full house or four of a kind. 1,292/1980 is what you should come up with. This is how many times you will miss. Subtract 1,292 from 1980 to get the number you need to find out how many times you catch a boat or quads. This comes out to 688/1980. You catch your cards to win every 688 times out of 1,980. Just divide the top number by two to come out to a close percentage. You will win the hand about 34% of the time.

Note that I didn't bother to calculate the times that the turn and river came 45 to give you two a split. To calculate getting cards on both the turn and river all you have to do is multiply the chances of the event occuring on the turn by the chances of the event occuring on the river. There will be a 4 or 5 come on turn 6/45 and the card need to give the board a straight 2 to 6 will come 3/44. So 18 times out of 1,980 you will split the pot. This gives you about one percent chance to split the pot.

So the answer is about 34% chance to scoop and about 1% to split the pot.

Answer 2

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Answer 3

1/8 of the time. For every 8 habds you will have a highly chance of getting a full house.