Distinct head-to-head match ups in holdem

Question

So I'm not very good at math, but I'm trying to improve.

I'm trying to understand this calculation that I read in Wikipedia, But I find it to be confusing, I would appreciate a clarification.

I understand why there are 169 distinct hands, but shouldn't there be 169*169 distinct match ups? Why isn't the 1225 opponent combinations are also distinct?

Thanks in advance.

Answer 1

Note 1 in the article on Hold'em Odds elaborates on this a bit further:

[Note 1] By removing reflection and applying aggressive search tree pruning, it is possible to reduce the number of unique head-to-head hand combinations from 207,025 to 47,008. Reflection eliminates redundant calculations by observing that given hands h_1 and h_2, if w_1 is the probability of h_1 beating h_2 in a showdown and s is the probability of h_1 splitting the pot with h_2, then the probability w_2 of h_2 beating h_1 is w_2 = 1 - (s + w_1), thus eliminating the need to evaluate h_2 against h_1. Pruning is possible, for example, by observing that Q♥ J♥ has the same chance of winning against both 8♦ 7♣ and 8♦ 7♠ (but not the same probability as against 8♥ 7♣ because sharing the heart affects the flush possibilities for each hand).

Your thinking was correct that 169x1225 doesn't make sense. The actual number is less than 169x1225, though not quite as small as 169x169. 169x278 ≈ 47,008.

With two cards there are only two unique 'suits' possible. Hands are either suited or off suited.

Adding two more cards gives more combinations of suits, now we can have the following suit possibilities:

1. 1111 - suited, suited, same suit
2. 1112 - suited, off suit, sharing suit
3. 1122 - suited, suited, different suits
4. 1123 - suited, off suit, different suits
5. 1223 - off suit, off suit, sharing one suit
6. 1212 - off suit, off suit, sharing both suits
7. 1234 - off suit, off suit, sharing no suits

Due to symmetry 1123 is the same as 2311 is the same as 3211; we ignore all symmetrical possibilities.

This doesn't fully explain the actual number, though it gets you most of the way there and gives a mental model of the possible combinations of suits with two hands.

Answer 2

How to get the 169

13 pair
78 unsuited 13 x 12 / 2
78 suited 13 x 12 / 2

Say I have a pair match up against unsuited
The unsuited can have
No matched to my suits
High card match to one of my suites
Low card match to one of my suites Both cards match to my suites

Pair to pair
13 * 1 for matching pair
13 * 12 * 3 for other pair
other pair can have 0, 1, or 2 suit matches
total 479