Where is my error?

[jmourning]

The FTR flop odds chart indicates the odds of flopping exactly one pair by matching a hole card = 26.939

I compute slightly different odds using the following math.

Can anyone tell me what my error is assuming the FTR chart is correct?
Starting with 2 unpaired cards, what is the probability that you will flop exactly 1 pair by matching a hole card?
Factors to consider:
There are 50 unseen cards.
Six of the unseen cards match one of the two cards in the starting hand.
Three of the unseen cards match one of the unpaired cards in the starting hand.
Three of the unseen cards match the other unpaired card in the starting hand.
Forty four of the unseen cards do not match either of the starting cards.
Using combinations:

Probability: (6 C 1)*(44 C 2) / (50 C 3) = 0.289591837

Odds ((1-0.2896))/0.2896=2.453

Using permutations:

Probability: (3 C 1) * (6 P 1)*(44 P 2) / (50 P 3) = 0.289591837

Odds ((1-0.2896))/0.2896=2.453

[Razvan729]

because other people are dealt cards also and there is the chance you dont have 6 outs to hit your pair on flop since they could be dealt to someone else. take into consideration this, do the math for 6outs/5outs/.../1outs/0outs, then combine these probabilities and do again the math for the 7 situations and you'll find the chance is ~26%.

[Sasquach991]

Poker Odds, Poker Probabilities, Common Flop Odds & Charts

Remember, it says EXACTLY.


If you scroll down it says:


"Holding unpaired cards and flopping EXACTLY one pair by matching a hole card.

I walked through that one above. You are looking for one of the 3 cards that match hole card 1 or the 3 cards that match hole card 2. So you are looking for one of 6 cards. Then you are looking for any card that DOES NOT match either of your hole cards (52 cards minus the 2 in your hand minus the 1 card that matched the first time minus the 5 cards that remain that match you hole cards = 44 cards). Then you are looking any card that DOES NOT match your hole cards AND does not match the card that just fell, because that would give you two pair (52 cards minus 2 in your hand minus the 1 card that matched the first time minus the 1 card that fell on the board minus the 8 cards that would match either your hole cards of the card that fell on the board = 40 cards.) Then divide by the total number of combinations, 50 cards * 49 cards * 48 cards. This comes to:

(6 * 44 * 40) / (50 * 49 * 48) = 8.97959%.

This is the chance of flopping one card to your hole cards ON THE FIRST CARD OF THE FLOP, as in:

Hole Cards:
Flop:

There is also that same chance to flop the pairing card on the second card, as in:

Flop:

And the same chance to flop the pairing card on the third card, as in:

Flop:

So you have 3 * 8.95959% or 26.939%

[jmourning]

Exceptionally clear reply. Best probability exposition I have received