Odds of a making a boat

[Brad22]

In Hold'em, when you have a pocket pair that makes trips on the flop, what are the odds of making a boat on the turn and river respectively?

I'll take a shot at this.

If you've made trips on the flop with a pocket pair, you need the board to pair to make a full house. You have two cards to do that with on the turn, with three remaining suits of each rank. Assuming no one else has these cards, that gives you six to make a full house with on the turn.

If there's one opponent (to keep the card counting simple), there'll be 45 cards left in the deck. (52 minus 4 hole cards minus the 3 cards on the flop.) On the turn, that'll mean you have 6/45 chance of making a full house on the turn, or 13%.

Using the same principles, if you don't make it on the turn, you'll have 9 cards out of 44 to pair the board on the river, or 20 percent.

Of course, these percentages could vary depending on the number of players and if one of them has made a pair with one of the board cards.

It's easiest to calculate this, if you calculate the odds of *not* pairing and then subtracting that from 100%.

We assume that the flop hasen't given you a full house, so the two extra cards on the board aren't paired.

The chance that the turn card misses these two cards is 41/47 (there are six cards that will pair the board).

After we've missed on the turn, the chance that the river will miss as well is 37/46.

So the chance of getting a full house on the turn or river is 1-(41/47)*(37/46) or 29.83%.

You might want to add the card that would give you quads as a great card, so the chance of getting a full house or quads is 1-(40/47)*(36/46) or 33.4%.