Odds of Two Pair on the Flop
It seems to happen at poignant moments to cause maximum irritation. You've just mucked a possibly playable hand and the flop suddenly shows two pair. An uneasy feeling that you've done something wrong comes over you. You were positive you did the right thing but now you can't but help feeling annoyed. Doesn't that just always seem to happen? But how often does this sort of thing occur?
Played Pockets
How often do you hit two pair on the flop? We'll put aside any case of pocket pairs since those are played differently. That leaves us with all the other combination's.
You'll get pocket pairs about 6% of the time, so 94% of all pockets will not be pocket pairs. See Odds of a Pocket Pair for details about this calculation.
Two on the flop
Now, how often does the flop match?
Once the two pocket cards are removed there are (50c3) = 19600 possible flops. Quite a lot of them are effectively the same in that the logical poker value is identical. That doesn't concern this calculation however. The question is how many contain the same two ranks as our pocket cards?
There are 3 more cards of the same rank as each pocket card. For example, let's say our pocket contains 2♥ 5♠. There may be 3 matching cards for each of these two cards in the deck so there are 9 possible ways that you can draw matching pairs on the flop, i.e. 2♠ 5♣, 2♠ 5♦, 2♠ 5♥, 2♣ 5♣, 2♣ 5♦, 2♣ 5♥, 2♦ 5♣, 2♦ 5♦, 2♦ 5♥. There are 44 remaining cards that could complete the flop (52 minus our pocket minus the 6 cards that can make two pair) so there are 9*44 = 396 flops that contain one of our 9 2-card combination's plus a third card that doesn't match either of our pocket cards. That gives us 396 / 19600 = 2.02% chance of flopping exactly two pair
In our calculation all hands resulting in a full house have been removed. This actually makes the calculation a lot simpler. If a full house and four of a kind is allowed, it'd add another 84 combination's.
Total
We know that 94% of the pockets aren't pocket pairs, so we apply that to our 2% chance of catching both of our pocket cards on the flop. 94% * 2% = 1.9%. So 1.9% is the over all likelihood of catching two pair on the flop when your pocket doesn't start with a pair in the first place, and not including the odds of catching a full house. That doesn't seem high! Another way to think of it is that this will happen about every 52nd hand. If full houses and quads are allowed that increases the odds to every 43rd hand.
Thus relying on the two pair means playing with around 50:1 odds. Such odds are never profitable in hold'em. Therefore whatever reason there was to fold the pocket was likely valid -- and the two pair coming up after the fact is completely irrelevant.
Permutation Method
An alternate way to calculate this is to use permutations. In this particular case it may actually be a shorter formula, but may be a bit more difficult to explain.
If one card is drawn there is a 6 in 50 chance to match against one of the pocket cards. If matched, there is now a 3 in 49 chance to match against the other pocket card. For the last card it must just not be another match, thus a 44 in 48 chance. So the chance to match the first and second card on the flop is 6/50 * 3/49 * 44/48 = 0.6735%.
Note that the final term 44/48 is required since only two pair interests us. If another matching card were to come up that would result in a full house, not two pair. Thus there are only 44 cards that are allowed.
There are three orders of the match: First, Second,Any; First, Any, Second; and Any, First, Second. Since we don't care which order the cards show up in, that means any of these three possibilities would be valid. 0.6735% *3 = 2.02% - the same probability we got using our earlier method.
Here's yet another slightly different way to get the same result: In this case the result for a single ordering is 3/50 * 3/49 * 44/48 = 0.3367%. In this case there are three terms: Match A; Match B; Any. That gives us (3*2*1) = 6 possible orderings for a total of 6 * 0.3367% = 2.02%.
Played Pockets
How often do you hit two pair on the flop? We'll put aside any case of pocket pairs since those are played differently. That leaves us with all the other combination's.
You'll get pocket pairs about 6% of the time, so 94% of all pockets will not be pocket pairs. See Odds of a Pocket Pair for details about this calculation.
Two on the flop
Now, how often does the flop match?
Once the two pocket cards are removed there are (50c3) = 19600 possible flops. Quite a lot of them are effectively the same in that the logical poker value is identical. That doesn't concern this calculation however. The question is how many contain the same two ranks as our pocket cards?
There are 3 more cards of the same rank as each pocket card. For example, let's say our pocket contains 2♥ 5♠. There may be 3 matching cards for each of these two cards in the deck so there are 9 possible ways that you can draw matching pairs on the flop, i.e. 2♠ 5♣, 2♠ 5♦, 2♠ 5♥, 2♣ 5♣, 2♣ 5♦, 2♣ 5♥, 2♦ 5♣, 2♦ 5♦, 2♦ 5♥. There are 44 remaining cards that could complete the flop (52 minus our pocket minus the 6 cards that can make two pair) so there are 9*44 = 396 flops that contain one of our 9 2-card combination's plus a third card that doesn't match either of our pocket cards. That gives us 396 / 19600 = 2.02% chance of flopping exactly two pair
In our calculation all hands resulting in a full house have been removed. This actually makes the calculation a lot simpler. If a full house and four of a kind is allowed, it'd add another 84 combination's.
Total
We know that 94% of the pockets aren't pocket pairs, so we apply that to our 2% chance of catching both of our pocket cards on the flop. 94% * 2% = 1.9%. So 1.9% is the over all likelihood of catching two pair on the flop when your pocket doesn't start with a pair in the first place, and not including the odds of catching a full house. That doesn't seem high! Another way to think of it is that this will happen about every 52nd hand. If full houses and quads are allowed that increases the odds to every 43rd hand.
Thus relying on the two pair means playing with around 50:1 odds. Such odds are never profitable in hold'em. Therefore whatever reason there was to fold the pocket was likely valid -- and the two pair coming up after the fact is completely irrelevant.
Permutation Method
An alternate way to calculate this is to use permutations. In this particular case it may actually be a shorter formula, but may be a bit more difficult to explain.
If one card is drawn there is a 6 in 50 chance to match against one of the pocket cards. If matched, there is now a 3 in 49 chance to match against the other pocket card. For the last card it must just not be another match, thus a 44 in 48 chance. So the chance to match the first and second card on the flop is 6/50 * 3/49 * 44/48 = 0.6735%.
Note that the final term 44/48 is required since only two pair interests us. If another matching card were to come up that would result in a full house, not two pair. Thus there are only 44 cards that are allowed.
There are three orders of the match: First, Second,Any; First, Any, Second; and Any, First, Second. Since we don't care which order the cards show up in, that means any of these three possibilities would be valid. 0.6735% *3 = 2.02% - the same probability we got using our earlier method.
Here's yet another slightly different way to get the same result: In this case the result for a single ordering is 3/50 * 3/49 * 44/48 = 0.3367%. In this case there are three terms: Match A; Match B; Any. That gives us (3*2*1) = 6 possible orderings for a total of 6 * 0.3367% = 2.02%.
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