Texas Hold 'Em possibilities

Question

Suppose you have been dealt “4♡, 5♡”.

Suppose you have one opponent. What is the conditional probability that you will win, given these two cards in hand, and that the board is “3♢, 4♢, 4♣, 4♠, 5♢”?

With this board, we have four of a kind. The only way the opponent will beat it is with a straight flush. How many possible two cards does the opponent have that can make a straight flush, regardless of order.?

What if you have two opponents? What is the conditional probability that you will win?

I am a little confused on these questions. Any help is appreciaited!

Answer 1

The maths is wrong... It should be 3 out of 1225 not 2 because 2d6d is possible as well as Ad2d and 6d7d.

But anyway the answer would be wrong for a real game - the answer would assume that the opponent had random cards, which isn't true in reality. It would depend on how the hand played out as to how likely an opponent is to have those cards. For example if it is 3-bet preflop hands like 6d7d are less likely; if it is a limped pot preflop they are more likely.

Answer 2

At the river there are 45 unknown cards
The board and your hole cards are known

Combin(45, 2) is 990

There are three possible hand that beat you

So 3/990 = 0.003030303 = 0.3030303% = 1 / 330

Against two opponents it does not double. It is complex as one can block the other.

Quads is a hand to play aggressively even with a straight flush draw on the board.

Answer 3

There are 45 × 44 ÷ 2 = 990 possible starting hands that can be dealt preflop to an opponent when you are heads up, discounting the known cards. (We divide by two previously because the order that the cards are dealt is irrelevant.)

In the situation that you have described above, he can have three of those starting hands, so the odds of a randomly dealt hand being one of those three would be 3 out of 990, which is 1:330, or 0.30% of the time.

I you have two opponents, the odds for the second opponent would be calculated in a similar manner, and you add the two together to determine the odds of either opponent having the hand.