I think your math is perfectly fine here:

If

**X=Q**
1st problem seems ok, we have 18 combos of PP's above & including QQ.

2nd problem, seems a bit strange.

We have six non-PP hands which are QQ+. (KQs, AQs, AKs, KQo, AQo, AKo)

12 combos for each off-suit hand + 4 combos for each suited hand = ( 3 x 12 )+( 3 x 4 ) = 48

# of hand combos above + including QQ+ = 66

Probability of being dealt a combo from this range = 66 / 1326 = 0.0498 =

**4.98% = approx 1/20 times**.

Let's say

** X=8** (

**Only** counting hands which are 88+, A8s+, K8s+, Q8s+, J8s+, T8s+, 98s, A8o+, K8o+, Q8o+, J8o+, T8o+, 98o)

So this exact range:

**# of PP:** 7 (8,8 - 9,9 - T,T - J,J - Q,Q - K,K - A,A)

**# of non-PP: ** 21 suited hands, each with 4 combos + 21 off-suit hands each with 12 combos (21 x 16 is ok)

Combos of PP's: 7 x 6 = 42

Combos of non-PP's: 21 x 16 = 336

Total combos: 42 + 336 = 378

Range = 88+, A8s+, K8s+, Q8s+, J8s+, T8s+, 98s, A8o+, K8o+, Q8o+, J8o+, T8o+, 98o

Probability of being dealt a combo from this range = 378 / 1326 = 0.285 =

**28.51%**
As you can see, your calculations are correct.

**The mistake** comes from the assumption that the above range accounts for 50% or more of your starting hands combos, which is not true. As you can see on the grid in the attached screenshot, it does not cover half of the total starting combos and therefore cannot be a correct assumption.

It's at this point, I recommend downloading our Equilab if you havn't done so already.

This is the download link.
It makes the visualization much more straightforward, in addition to this, it lets us do the calculation in a different way.

We want to calculate the top 50% of our starting hand range? (As you want to know the 50% of the combos on the top-side of the range grid so we must go from here)

We know that there are 1326 combos of starting hands, so the range we want to find includes 1326 / 2 = 663 starting hand combos.

Suited cards in general, account for a smaller percentage of the top of our range than off-suit cards because they have less combos, pocket pairs account for the lowest percentage. Because the top of our range includes more suited cards than off-suit cards, the bottom of our range is perceived (in our minds or on the grid) to be physically larger than it actually is in terms of the starting hand itself.

Here is the grid again, with the top 50% of our range, excluding all PP's for purposes of a more accurate perception.

As you can see, Off-suit hands account for less of the grid than the suited hands, but there are actually more off-suit combos of course. This image is the same in our minds when we think "There are 13 cards, so every card valued about 7 must be more than 50% of the hands."

There are two things we need to note about this thought process:

1. It's incorrect

2. We are trying to define the top 50% of our range, simply because we want to know the 50% of the cards with the highest values. (Remember we also have the bottom of our range)

*Why is it incorrect?*
It's incorrect because we make the assumption that each hand has the same amount of combos to be dealt within a defined range, on the back of the assumption that we are not taking suits into account, as we are only using numbers. This skews our perception of the starting hands possibilities and should be avoided.

I hope this helps, this stuff can be confusing at times. Always think in combos!

Now, We need to determine what we want to find out:

*Do we want to know if this range*** (88+, A8s+, K8s+, Q8s+, J8s+, T8s+, 98s, A8o+, K8o+, Q8o+, J8o+, T8o+, 98o)** accounts for 50% or more of our starting hand possibilities?
Simple answer is no, it includes 28.51% of our starting hand possibilities, as this particular range, has 378 combos, which is nowhere near 50% x 1326.

*Do we want to know which hands are included in the top 50% of our range & why these hands represent the top 50%?*
Here is the top 50% of our range:

**33+, A2s+, K2s+, Q2s+, J4s+, T6s+, 96s+, 86s+, 76s, 65s, A2o+, K5o+, Q7o+, J7o+, T7o+, 98o**
*Why do these hands represent 50% of our range?*
Because these are the 50% of combos of starting hands which will hold the most value in the long-term. Keep in mind that we shouldn't play anywhere near 50% of our starting hands, which is why you see trash (K5o, J7o, etc..) hands in the above range I gave.

*Do we want to know the number of combos of hands which are included in the top 50% of our range?*
That's easy it's 50% of 1326 = 663 combos

Just a quick thing on the back of the last question, we cant have 663 (50%) combos in our range, we need to have 660 (49.77%) or 664 (50.08%). Your number of combos will always be an even number as no starting hand has less than four combos + are all multiples of two. We can only have an odd number if we include a

**single** specific combo,

There is one combination of 6

5

, but 12 combos of 65o + 4 combos of 65s, hence a defined range would only include an odd number of combos if we ourselves include a specific starting hand AND the suits for each card.

I hope this helps, let me know if you have any other questions.

-E