Beginner Poker Math - Homework

[Erpel]

THIS POST CONTAINS BASIC POKER MATH EXERCISES FOR BEGINNERS TO COMPLETE. IF YOU WANT TO TRY TO FIGURE THEM OUT FOR YOURSELF DO NOT READ PAST THIS INITIAL POST.

The exercises here are simple and some questions are along the lines of asking what colour the sky is (blue) simply to make the person reading the exercises in numerical order think about the factors that they need to focus on going into the next question and not because it's a meaningful answer in itself.

Situation:
$0.10/$0.25 blinds, 9 man table:
Preflop: Villain raises UTG to $1, you call on the button with 8d6d and the blinds fold
Flop, pot is $2.35, 2 players: Tc5d2d, villain raises to $2, you call
Turn, pot is $6.35, 2 players: 7s, villain goes all-in for $6

The bad play:
Calling preflop was bad here in every sense of the word because stacks are too short to play this hand profitably (villain has $9 which is 36bb). But at least we hit the flop as well as could be hoped. Don't judge the preflop play - just setting up a calculation exercise.

Villain:
His stats are something like 40/4/1 but to make our math simple let's say we know him much better than that. Couple of villain profiles:
A) PF raise is only done with AA, KK, QQ, JJ, AK, AQ and done with these hands 100% of the time. cbets flop 100% of the time. Shoves turn 100% of the time.
B) PF raise is only done with AA, KK, QQ, JJ, AK, AQ and done with these hands 100% of the time. cbets flop 100% of the time. Shoves turn only with AA, KK, QQ, JJ and 100% of the time.
C) PF raise is only done with QQ, JJ, AK, AQ (limp/re-raises AA, KK) 100% of the time. cbets flop 100% of the time. Shoves turn 100% of the time.
D) PF raise is only done with QQ, JJ, AK, AQ (limp/re-raises AA, KK) 100% of the time. cbets flop 100% of the time. Shoves turn 100% only with QQ, JJ, AdKd, AdQd

Note that in the villain profiles above I have emphasised that he plays specific pocket holdings the same way EVERY time. No limping, no slowplaying, no plays based on mood or flow or anything like that. The reason I've chosen this assumption is that it makes hand combinations exactly accurate when it comes to doing equity calculations.

Note further that it's not important to strive for exact numbers here. I don't care if you calculate the equity needed with 17 digits or use an odds notation and abbreviating 31 to 15 to 2 to 1. What's important for these exercises is PICKING THE RIGHT NUMBERS FOR WHAT YOU WANT TO DETERMINE AND COMBINING THEM IN A WAY THAT ANSWERS THE QUESTION.

Questions:
1) On the turn, how much do you have to call to win how much? On the turn, what are your pot odds?
2) On the turn, how large a percentage of the time do you need to win the hand for calling to be profitable (same question as 1) just using % notation)
3) Given profile A-B, how many hand combinations are possible for each of the villain's starting hands?
4) Given profile A-B, on the flop, how many hand combinations are overpairs or top set, and how many hand combinations are two unpaired overcards? Express overpair / top set as a percentage of his range.
5) Given profile C-D, how many hand combinations are possible for each of the villain's starting hands?
6) Given profile C-D, on the flop, how many hand combinations are overpairs or top set, and how many hand combinations are two unpaired overcards? Express overpair / top set as a percentage of his range.
7) On the turn, how many outs do you have against AA, KK, QQ, JJ?
8) On the turn, how many outs do you have against AK, AQ if they are not AdKd or AdQd?
9) On the turn, how many outs do you have against AdKd, AdQd?
10) How many hand combinations are AK, AQ when not AdKd, AdQd?
11) How many hand combinations are AdKd, AdQd?
12) Given a hand being in the AA, KK, QQ, JJ range - what is the expected value of calling the turn shove?
13) Given a hand being in the AK, AQ (no AdKd, no AdQd) range - what is the expected value of calling the turn shove?
14) Given a hand being in the AdKd, AdQd range - what is the expected value of calling the turn shove?
15) Combine the results from 12-14 based on the number of hand combinations to give you an expected value of calling the turn shove with all of range A
16) Combine the results from 12-14 based on the number of hand combinations to give you an expected value of calling the turn shove with all of range B
17) How many hand combinations are QQ, JJ and given a hand being in the QQ, JJ range - what is the expected value of calling the turn shove?
18) Combine the results from 12-14 + 17 based on the number of hand combinations to give you an expected value of calling the turn shove with all of range C
18) Combine the results from 12-14 + 17 based on the number of hand combinations to give you an expected value of calling the turn shove with all of range D

For the flop assume that we are doing two streets of cards, but only one street of betting. By this I mean that AA, KK, QQ, JJ will stack off in any case, and AK, AQ will either stack off, or check and FOLD on the turn so you win no more money from them. AdKd and AdQd will also stack off.

These flop exercises revolve around implied odds, which is value extracted on a later street if you decide to continue in the hand. In the present case we have enough knowledge that we can make a complete expected value calculation and state exactly against which of villain ranges A-D we should call or fold and what the expected value of each is. But I will focus on implied odds calculations here.

For the below calculations feel free to ignore the following
- If a pair or unpaired hand contains one diamond the board can go runner, runner diamonds and have you lose to a better flush. Just assume a flush card hitting means you win.
- Your runner runner straight draw (which became strong in the present case through the 7s coming on the turn) - just count the flush outs on the flop, but consider them clean outs to win unless up against AdKd, AdQd.
- Ignore that if you hit your pair to beat AK / AQ you may still be outdrawn by an A, K or Q on the river.

19) On the flop, how much do you have to call to win how much if the opponents hand is in stack-off range? Use pot odds or % notation as you prefer.
20) On the flop, how much do you have to call to win how much if the opponents hand is in non-stack-off range? Use pot odds or % notation as you prefer.
21) Combine hand combinations for the ranges with outs for the subranges to approximate a single number of outs for a whole range. Example - 40% 6 outs, 40% 10 outs, 20% 2 outs is 6.8 outs for the range. Do this for all of ranges A-D
22) Does the implied odds justify calling the flop against all ranges A-D? (Combine 19-21)
21) Assume 9 clean outs and a villain hand in the stack-off range - how much would the villain need to bet you for a call to be break-even based on implied odds?

[Unibomber14]

Oooh, is there a full poker math workbook on the market, or is that idea still up for grabs?

[settecba]

GREAT POST

i´ll give it a try later when i find time to do it. I´ll probably make tons of mistakes, i dont care. I´d really like to know where im wrong and why.

after i do it, should i post my answer here? or pm?

[OhBollocks]

GREAT POST

Seconded. Post reply after work tomorrow. Thanks man

[Erpel]

I haven't actually done any of the calculations, just eyeballed the situation for what I wanted to include. I did think of including TT in the range and have the Td be not a flush out because it's a quad out etc, but I figured AdKd etc would cover off the dirty out dimension.

I'll leave it on here for a few days then I'll calculate it from scratch myself and post the results - and I'll comment on anyones results as well.

[settecba]

ok..these are my answers...probably with some mistakes, id like you to point them out. didnt include the calculations so everyone can do it from scratch without any "influence", if you want them i can post them.

had fun doing it...TY ERPEL

1)2.06
2)32.70%
3)24 AA;KK;QQ;JJ (6 of each)
32 AK;AQ (16 of each)
4)24 overpairs (42.86%)
32 unpaired overcards
5)12 QQ;JJ (6 of each)
32 AK;AQ (16 of each)
6)12 overpairs (27.27%)
32 unpaired overcards
7)15
8)15
9)12
10)30
11)2
12)-0.016304348
13)-0.016304348
14)-1.213043478
15)-0.059045031
16)-0.016304348
17)12 QQ;JJ combinations. EV:-0.016304348
18.1)-0.070701581
18.2)-0.187267081
19)8.35:2 or 4.175:1
20)4.35:2 or 2.175:1
21)A and B: 12.10714286
C and D:12.95454545
22)A and C: YES
B and D: NO
23)$4.03

[daven]

I'll leave it on here for a few days then I'll calculate it from scratch myself and post the results - and I'll comment on anyones results as well.

cool, i;ll leave it a few days longer than that and comment on your results as well

[overflow]

I misread the original post so I'm using $12.25 and not $12.35.

1) We must call $6 to win $12.25 or 2.04:1
2) >= 33%
3) 56 combinations ( 4x6 combinations of pairs, As, Ks, Qs, Js, 2x16 combinations of suited and unsuited non-paired cards AK and AQ)
4) 24/56 are overpairs, top set is not in his range. If 24 combinations are overpairs 32/56 combinations are unpaired hole cards. 42% of his range is overpairs.
5) There are 44 combinations of hands in his range. 6 combinations of JJ, 6 combinations of QQ, 16 combinations each of AK and AQ.
6) 12/44 are overpairs, top set is not in his range. if 12 combinations are overpairs 32 combinations are unpaired overs. 27.2% of his range is overpairs.
7) 15 outs, 9 diamonds, any 9, and any 4 (not diamonds).
8) 21 outs, 9 diamonds, any 9, any 4, any 8, and any 6
9) 12 outs, any 8, and any 6, any 4 or 9 that's not a diamond.
10) There are 15 combinations of AK or AQ that are not specifically the Ad{K,Q}d, 30/44 a bit more that 68.2% of his range.
11) There are 2 combinations of AdKd and AdQd, 1 each. This comprises 2/44 or just under 5% of his range.
12) (15/46)(12/24)(12.25) + (14/46)(12/24)(12.25) + (31/46)(12/24)(-6) + (32/46)(12/24)(-6) = -.2473
13) (20/46)(12/24)(12.25) + (19/46)(12/24)(12.25) + (26/46)(12/24)(-6) + (27/46)(12/24)(-6) = 1.7364
14) (12/46)(12.25) + (34/46)(-6) = -1.2391
15) This is complicated, because sometimes he's counterfeiting your outs. 3 combinations of each pair contain a counterfeiter.
But AK and AQ are also in his range so 12 more combinations are counterfeiting your outs. So 24/56 combinations counterfeit
a single flush out. The expected value calculation would be:
(15/46)(24/56)(12/24)(12.25) + (14/46)(24/56)(12/24) + (21/46)(30/56)(18/30)(12.25) + (20/46)(30/56)(12/30)(12.25) + (12/46)(2/56)(12.25) +
(31/46)(24/56)(12/24)(-6) + (32/46)(24/56)(12/24)(-6) + (25/46)(30/56)(18/30)(-6) + (26/46)(30/56)(12/30)(-6) + (34/46)(2/56)(-6) = $0.2901 +EV
16) (15/46)(12/24)(12.25) + (14/46)(12/24)(12.25) + (31/46)(12/24)(-6) + (32/46)(12/24)(-6) = $0.2473 -EV
17) 12 combinations. (15/46)(6/12)(12.25) + (14/46)(6/12)(12.25) + (31/46)(6/12)(-6) + (32/46)(6/12)(-6) = $0.2473 -EV
18a) (15/46)(12/44)(6/12)(12.25) + (14/46)(12/44)(6/12) + (20/46)(30/44)(18/30)(12.25) + (19/46)(30/44)(12/30)(12.25) + (12/46)(2/44)(12.25) +
(31/46)(12/44)(6/12)(-6) + (32/46)(12/44)(6/12)(-6) + (26/46)(30/44)(18/30)(-6) + (27/46)(30/44)(12/30)(-6) + (34/46)(2/44)(-6) = $0.6202 +EV
18b) (15/46)(12/14)(6/12)(12.25) + (14/46)(12/14)(6/12)(12.25) + (12/46)(2/14)(12.25) + (31/46)(12/14)(6/12)(-6) + (32/46)(12/14)(6/12)(-6) + (34/46)(2/14)(-6) = $0.3890 -EV
19) You have to call $2 to win 10.25 or 5 1/8 : 1 or about 5:1 you need a little under 20% equity.
20) You have to call $2 to win a minimum of $4.35. Direct odds are 4.25:2 or 2 1/8 : 1 you have to have at least 30% equity or more.
If villain will either go all-in or fold to any bet, and his hand is in the non-stackoff range our odds are 2.125:1 we need 32% equity.
21a) 24 of 56 combinations are pairs 12 of them have one of our outs counterfeited (12/56)(9) * (12/56)(8) * (18/56)(15) * (12/56)(14) * (2/56)(6) = ~10.2 outs for the range
21b) Same as A. his range on the flop is the same for A and B, 10.2 outs for the range.
21c) 12 of 44 combinations are pairs, 6 counterfeit a flush out, (6/44)(9) * (6/44)(8) * (18/44)(15) * (12/44)(14) * (2/44)(6) = 8.6 outs for the range
21d) Same as C. his range on the flop is the same for C and D, 8.6 outs for the range
22a) (10/47)+(10/46) 43% equity against villains range, it's an easy call, as we only need 20% equity.
22b) He shoves 26/56 or 46.52% and folds 53.48%. (26/56)((10/47)+(10/46)) + (30/56) = 73.54% equity, easy call because of fold equity.
22c) (8.5/47)+(8.5/46) 36.56% equity against villain's range, again easy call as we only need 20%
22d) 14 out of 44 combinations shove 30 out of 44 fold. (14/44)(.3656) + (30/40) = 86.63% equity against villain's range.
22x) I believe we can profitably call in all scenarios A,B,C, and D.
23a) $7.74 assuming we have 43% equity on the flop.
23b) If we have 73.5% equity (due to fold equity) in this scenario, villain can not bet enough to take away our implied odds (and avoid becoming pot committed in such a way as to violate the contraints of scenario B)
23c) $5.89 assuming we have 36.56% equity.
24d) Same as B, villain can't bet enough without pot committing himself in a manner that violates the exercise.

[settecba]

question 8 is 21 outs!!! dont know what i was thinking, a lot of my other answers are wrong based on that. I also didnt calculate EV taking into account the hands where our opp has 1 of our outs, is the difference in the results significant because of this?

[overflow]

question 8 is 21 outs!!! dont know what i was thinking, a lot of my other answers are wrong based on that. I also didnt calculate EV taking into account the hands where our opp has 1 of our outs, is the difference in the results significant because of this?

He tells you to ignore the effect of counterfeited outs and villain redraws, I like being a bit more accurate so I decided to compensate for it. I'm sure it's a lot easier if you don't do this I don't believe it will significantly alter your expectancy calculations as the margin of error you're introducing by neglecting 1 out twice is about 4%.

[wellrounded08]

Could one of you fella's hook me up with a link on how to calculate EV? FE and Equity in general? I know right? "what a fish! LOL" But it's ok I'll learn.

[overflow]

EV is easy. Multiply the frequency at which an event occurs (I hit a clean 9 out draw with two cards to come 38.7% of the time and I'll make my flush (9/47 + 9/46)) by the return you expect to receive (e.g. the size of the pot), then take the % of the time you expect to lose and multiply it by the size of the bet you're making or calling. Subtract this result from the original result and you have the net expected value for the decision.

10NL 6max - Villain is a standard 45/5/1.0 retard.

SB ($10)
BB($10)
Hero($10)

Hero is on the button and is dealt [QsTs]
3 folds, Hero raises to $0.35, small blind calls, big blind calls.

Pot is $1.05 the flop is [ 3s8cAs ]
The small blind bets $0.80, making the pot $1.85
big blind folds
The pot is laying us 1.85/0.80 or roughly 2.3:1
Our equity to see the turn card is 9/47 or 19.14%
and we need a little over 5 : 1 to profitably call.
If we were to call the expected value calculation for the single street of the flop would be:

   A     B       C       D
(9/47)(1.85) + (38/47)(-0.80) = $0.29 -EV

A) The frequency at which you expect to make your flush
B) The expected return if you make your hand
C) The frequency at which you expect to miss your flush
D) The amount you lose if you miss

You lose an average of $0.29 (or 2.9bb) everytime you make this call and villain check/folds the turn with 100% of his range (this is unfeasible but it simplifies the math) and villain bets any non-spade turn and we fold.

This is definitely an oversimplification, but it's where you need to start.

Let's add an interesting twist though, let's say that we'll stack him 35% (random percentage) if we hit on the turn, and we'll fold to a bet if we miss, and he'll always bet a non-spade turn.

Now our calculation has changed a bit:
(9/47)(.35)(11.95) + (9/47)(.65)(1.85) + (38/47)(-0.80) = $0.38 +EV
or a positive expectancy of 3.8bb every time this situation occurs.

In 9 out of 47 cases, 35% of the time we stack him, and 65% of the time he check folds. In 38 out of 47 cases, we'll miss, he'll bet and we'll fold losing our $0.80 investment.

This is where things like fold equity come into play (from an arithmetical standpoint), there is a certain percentage of the time where your opponent will make different actions based on his range. So you apply probabilities for his action given each holding (What Erpel referred to as 'subrange') and apply them individually to the probability scenarios for the outcome of your own hand.

The easy way to show an example of fold equity calculation is on the river.

10NL 6max 100bb stacks
Hero is dealt [7d6d]

The board reads:
[As Kc Ts 5d Qh]

You raised preflop to 4x and were called by only the big blind. The big blind checks, you bet .75 into .85, and the big blind calls. The turn goes check/check, and the big blind checks to you on the river.

Our hand has almost no showdown value whatsoever here. If we want to win this pot we have to bet. Let's say we bet 1.80 into a 2.35 pot. If we expect our villain will fold all holdings that aren't two pair or better (hypothetically) which comprises about 85% or more (we'll use 85 as a round number) of his range.

The expected value calculation for this scenario is:
(.85)(2.35) + (.15)(-1.80) = $1.73 +EV

If villain folds 85% of his range to a $1.80 bet on the river, we make an average of 17.3bb every time we make this play. That's how fold equity works.

Now the trick to putting it all together is learning how to incorporate fold equity and implied equity calculations for future streets into your expected value calculations for the street you're evaluating to make a better decision about what to do now. These are, essentially, the raw components of hand planning. (A is the current size of the pot, x(1,2,3) is the percentage of the time a card will come that will give me fold equity (villain folds), implied equity(villain calls a future bet when we have the best hand), or null equity (we have to fold) respectively, the percentage of fold equity will be y, the future bet you intended to make will be b, and my implied equity (how frequently we expect a bet to get paid off on a future street) based on the size of villain's range will be z. The amount we're betting or calling on the flop will be c.

The EV formula would be:
(x1)(y)(A) + (x2)(z)(A+b) + (x3)(-c)
or
(x1)(y)(A) + (x2)(z)(A+b) - (x3)(c)

In plain english this reads: X1 percent of the time a card will come that will cause my opponent to fold to my bet y percent of the time, and yield A (the pot size) in profit plus X2 percent of the time a card will come that will cause my opponent to call a bet of size 'b' 'z' percent of the time, resulting in a total profit of A+b subtracting for the X3 percent of the time a card comes that will cause my opponent to bet, and I will fold.

In order for this calculation to work y and z must have a sum of 1, otherwise there will be cases in which we are betting the turn with the worst hand that need to be accounted for.

In summary:
Frequency * Expectancy = Total Expected Value.

You can have any number of elements in the EV calculation, just make sure all the case scenarios add up to 100%. In this example X1 + X2 + X3 should equal 1, and you're properly subtracting for all scenarios that result in one or more lost bets.

[OhBollocks]

Okay, I didnt read any of the answer replies. I know this is just basic stuff but it still has me thinking properly.Cheers man

1) Call $6 to win $12.35. Pot odds are 2.06:1
2)33%
3)56
4)Overpairs/top set =24......Overcards =32.........overpairs 42%
5)44
6)Overpair/top set=12......Overcards=32........overpairs 28%
7)15
8)21
9)12 (cant double count 4d,9d)
10)30
11)2

I'll finish the Qs in an edit 2morrow

[wellrounded08]

Thanks alot Overflow. Seriously, that was alot of help. Wow, maybe we can expect a whole poker workbook.
I read through it in its intirety, slowly, changed around some figures to look at different results and how different percentages altered the EV, and will probably reread it in the near future.

[settecba]

question 8 is 21 outs!!! dont know what i was thinking, a lot of my other answers are wrong based on that. I also didnt calculate EV taking into account the hands where our opp has 1 of our outs, is the difference in the results significant because of this?

He tells you to ignore the effect of counterfeited outs and villain redraws, I like being a bit more accurate so I decided to compensate for it. I'm sure it's a lot easier if you don't do this I don't believe it will significantly alter your expectancy calculations as the margin of error you're introducing by neglecting 1 out twice is about 4%.

TY overflow

[Erpel]

1) On the turn, how much do you have to call to win how much? On the turn, what are your pot odds?
You call $6 to win $12.35. This makes your pot odds 12.35 to 6 or 2.06 to 1.

2) On the turn, how large a percentage of the time do you need to win the hand for calling to be profitable (same question as 1) just using % notation)
If you call $6 the pot will be $18.35 and you will need to win 6/18.35 = 32.7% of the time to be breakeven. Win more and you are profitable, win less and you are not profitable.

3) Given profile A-B, how many hand combinations are possible for each of the villain's starting hands?
6 (AA) + 6 (KK) + 6 (QQ) + 6 (JJ) + 16 (AK) + 16 (AQ) = 56

4) Given profile A-B, on the flop, how many hand combinations are overpairs or top set, and how many hand combinations are two unpaired overcards? Express overpair / top set as a percentage of his range.
Overpair: 24 (AA/KK/QQ/JJ) | Unpaired overcards: 32 (AK, AQ) | Percentage of overpairs: 24 / 56 = 42.9%

5) Given profile C-D, how many hand combinations are possible for each of the villain's starting hands?
6 (QQ) + 6 (JJ) + 16 (AK) + 16 (AQ) = 44

6) Given profile C-D, on the flop, how many hand combinations are overpairs or top set, and how many hand combinations are two unpaired overcards? Express overpair / top set as a percentage of his range.
Overpair: 12 (QQ/JJ) | Unpaired overcards: 32 (AK, AQ) | Percentage of overpairs: 12 / 44 = 27.3%

7) On the turn, how many outs do you have against AA, KK, QQ, JJ?
9 (all diamonds) + 3 (9s/9c/9h) + 3 (4s/4c/4h) = 15

8) On the turn, how many outs do you have against AK, AQ if they are not AdKd or AdQd?
9 (all diamonds) + 3 (9s/9c/9h) + 3 (4s/4c/4h) + 3 (8s/8c/8h) + 3 (6s/6c/6h) = 21

9) On the turn, how many outs do you have against AdKd, AdQd?
3 (9s/9c/9h) + 3 (4s/4c/4h) + 3 (8s/8c/8h) + 3 (6s/6c/6h) = 12

10) How many hand combinations are AK, AQ when not AdKd, AdQd?
16 (AK) + 16 (AQ) - 1 (AdKd) - 1 (AdQd) = 30

11) How many hand combinations are AdKd, AdQd?
1 (AdKd) + 1 (AdQd) = 2

12) Given a hand being in the AA, KK, QQ, JJ range - what is the expected value of calling the turn shove?
Method 1: (15/46) * $12.35 - (31/46) * $6 = $4.0272 - $4.0435 = -$0.0163
Method 2: (15/46) * $18.35 - $6 = -$0.0163
Method 1 says - I win what's in the pot when I win, but I lose my bet when I lose
Method 2 says - I put money in the pot regardless - that's what I pay. When I win, I win what's there now and what I paid also
Both work - question of taste.

13) Given a hand being in the AK, AQ (no AdKd, no AdQd) range - what is the expected value of calling the turn shove?
Method 1: (21/46) * $12.35 - (25/46) * $6 = $5.6380 - $3.2609 = +$2.3772
Method 2: (21/46) * 18.35 - $6 = +$2.3772

14) Given a hand being in the AdKd, AdQd range - what is the expected value of calling the turn shove?
Method 1: (12/46) * $12.35 - (34/46) * $6 = $3.2217 - $4.4348 = -$1.2130
Method 2: (12/46) * $18.35 - $6 = -$1.2130

15) Combine the results from 12-14 based on the number of hand combinations to give you an expected value of calling the turn shove with all of range A
Using numbers as truncated above to 4 digits after the decimal symbol - some rounding errors possible on the third and fourth digit.
(24 * -$0.0163 + 30 * $2.3772 + 2 * -$1.2130) / 56 = +$1.2232

16) Combine the results from 12-14 based on the number of hand combinations to give you an expected value of calling the turn shove with all of range B
When I corrected range D to include AdKd and AdQd I should have added them to range B also, but since I didn't - range B is simple:
(24 * -$0.0163) / 24 = -$0.0163

17) How many hand combinations are QQ, JJ and given a hand being in the QQ, JJ range - what is the expected value of calling the turn shove?
Hand combinations: 6 (QQ) + 6 (JJ) = 12
EV - same as 12): -$0.0163

18) Combine the results from 12-14 + 17 based on the number of hand combinations to give you an expected value of calling the turn shove with all of range C
(12 * -$0.0163 + 30 * $2.3772 + 2 * -$1.2130) / 44 = +$1.5612

18) Combine the results from 12-14 + 17 based on the number of hand combinations to give you an expected value of calling the turn shove with all of range D
(12 * -$0.0163 + 2 * -$1.2130) / 14 = -$0.1873

On review I am completely unhappy with how I stated the implied odds questions. I should have created a completely different hand for it (or just changed the stack sizes tbh) as the way these questions are posed don't actually lend themselves to the simplistic calculations I wanted to do.
The questions really are rubbish so I'll be somewhat brief in my answers.

19) On the flop, how much do you have to call to win how much if the opponents hand is in stack-off range? Use pot odds or % notation as you prefer.
On the flop I call $2 to win a pot of $4.35 as well as the $6 behind. Since this question specifies that it relates to the stackoff range the assumption is that I hit a hand on the turn (9 or 15 outs depending on which part of which range) I will be able to get the rest of the money by calling an AI.
Pot odds notation: 10.35 / 2 - 1 = 5.175 => Pot odds are 5.175 to 1.
Percentage equity notation: 1 / 6.175 = 16.19% - if I win more than 16.19% of the time it's profitable to call. For the purpose of this implied odds calculation I define winning as hitting a hand on the turn that I rate to be best.

20) On the flop, how much do you have to call to win how much if the opponents hand is in non-stack-off range? Use pot odds or % notation as you prefer.
If he's in non-stackoff range I'd defined that he'll auto-fold to any turn bet meaning we don't care what the turn card is - we always win.

21) Combine hand combinations for the ranges with outs for the subranges to approximate a single number of outs for a whole range. Example - 40% 6 outs, 40% 10 outs, 20% 2 outs is 6.8 outs for the range. Do this for all of ranges A-D
My assumption is a hand that rates to be best on the turn. I'm again going with a B range that (stupidly) folds AdKd, AdQd even if it hit the flush to be consistent with the posted ranges.
Against AA, KK, QQ, JJ that means I hit my flush (9 outs)
Against AK, AQ (no AdKd, AdQd) that means I hit my flush or a pair (15 outs)
Against AdKd, AdQd that means I hit my pair, no flush (6 outs)
Range A (stackoff): (24 * 9 + 30 * 15 + 2 * 6) / 56 = 12.11 outs average for 56 hand combinations
Range B (stackoff): (24 * 9) / 24 = 9 outs average for 24 hand combinations
Range B (non-stackoff): (30 * 15 + 2 * 6 outs) / 32 = 14.44 outs average for 32 hand combinations (or just 47 outs as he'll always fold)
Range C (stackoff): (12 * 9 + 30 * 15 + 2 * 6) / 44 = 12.95 outs average for 44 hand combinations
Range D (stackoff): (12 * 9 + 2 * 6) / 14 = 8.57 outs average for 14 hand combinations
Range D (non-stackoff): (30 * 15) / 30 = 15 outs average for 30 hand combinations (or just 47 outs as he'll always fold)

22) Does the implied odds justify calling the flop against all ranges A-D? (Combine 19-21)
Range A - stackoff (all):
Yes - odds are (47 - 12.11) / 12.11 = 2.88 to 1 against winning and the 5.175 to 1 offered by the pot makes this profitable.
Yes - we win 25.77% of the time which is more than the 16.19% required to be profitable

Range B - stackoff (24 hand combinations):
Odds are: (47 - 9) / 9 = 4.22 to 1 against (profitable)
Percentage equity: 9 / 47 = 19.15% (profitable)
Range B - non-stackoff (32 hand combinations):
He'll always fold - all are profitable. Range B is a profitable call.

Range C - stackoff (all):
Yes - odds are (47 - 12.95) / 12.95 = 2.63 to 1 against (profitable)
Percentage equity: 12.95 / 47 = 27.55% (profitable)

Range D - stackoff (14 hand combinations):
Odds are: (47 - 8.57) / 8.57 = 4.48 to 1 against (profitable)
Percentage Equity: 8.57 / 47 = 18.23% (profitable)
Range D - non-stackoff (30 combinations):
He'll always fold - all are profitable. Range D is a profitable call.

21) Assume 9 clean outs and a villain hand in the stack-off range - how much would the villain need to bet you for a call to be break-even based on implied odds?
In this question I am assuming we're on the flop with two cards to come and 9 clean outs - meaning any of those 9 cards coming on either turn or river means we win regardless what other cards fall (we cannot be outdrawn). Further the assumption is that the turn checks through regardless and any river bet we make will be called. And if we do not hit one of our 9 outs we will 1) not put any more money in and 2) lose.
Chance to win - to make hand on either turn or river: 1 - (38 / 47 * 37 / 46)
Chance to win is 34.97%
Chance to lose is 65.03%

Method 1:
EV: 34.97% * ($18.35 - $x) - 65.03% * $x
We are looking for the breakeven - the breakeven is when EV is = 0.

34.97% * (18.35 - x) - 65.03% * x = 0 =>
34.97% * 18.35 - 34.97% * x = 65.03% * x =>
34.97% * 18.35 = (65.03% + 34.97%) * x =>
6.4170 = 1 * x =>
x = 6.4170

Method 2:
EV: 34.97% * 18.35 - $x

34.97% * 18.35 - x = 0 =>
x = 34.97% * 18.35 = 6.4170

Thus, if villain bets $6.4170 into the $2.35 alread in the pot we are breakeven to call for implied odds.

Other implied odds shenanigans:
The pot is $2.35. The bet to me is $2.
I have 10 outs and we're on the flop. That means 10 cards help me, 37 cards do not help, my odds of hitting my hand are 3.7 to 1 against.
I take this 3.7 and use it as a multiplier on the bet. $2 * 3.7 is $7.4. I need to WIN $7.4 in order for me CALLING $2 to be profitable. The current pot is $4.35, so I will need to win $3.05 on later streets on average for calling to be profitable. After I call the pot will be $6.35, so I will need a half PSB to be called on average to be profitable - when I hit my hand.