|
1. How often does Villain have to fold for Hero to have a +EV bluff?
This is just bet/(bet+pot) which here is just 1/(1+P). If you don't understand this, stop reading this thread and go study my mathematics of EV thread.
2. If Villain calls with a 7 or higher, what hands can Hero make a +EV value bet with?
Look at his calling range:
A K Q J T 9 8 7
We can only bet hands that will beat more hands than it loses to. The worst hand that meets this requirement is a Jack which beats 4 and loses to 3 of the hands in Villain's calling range. Therefore, we can make a +EV value bet with an Ace, King, Queen or Jack.
3. If each ante is 0.25 units, what is the minimal number of hands Villain has to call a bet with so that Hero can't make a +EV bluff?
If each ante is 0.25 units then the pot is 0.5 units. We're making a bet of 1 unit, so we need our opponent to fold bet/(bet+pot) percent of the time, which here is 1/(1+0.5) = 1/1.5 = 66.7%. So we need Villain to fold 66.7% of the 12 hands in his range, thus calling with at least 4 hands, so he'll have to call with at least a Jack or better to keep Hero from making +EV bluffs.
4. If Villain calls with an 8 or higher, what is the EV of betting a T for Hero? of checking a T for Hero?
First the EV of betting. There are 4 out of 12 cards (A, K, Q, J) that Villain will call with and we lose 1 unit. Then there are 2 out of 12 cards (9, 8) that Villain will call with and we win the pot of P plus the 1 unit he calls. Finally, there are 6 out of 12 cards (7, 6, 5, 4, 3, 2) that Villain will fold and we collect the pot of P. That makes our EV of betting:
(4/12)(-1) + (2/12)(P+1) + (6/12)(P)
Now the EV of checking. There are 4 out of 12 cards (A, K, Q, J) that we will lose to at showdown and show a profit of 0 on our check. Then there are 8 out of 12 cards (9, 8, 7, 6, 5, 4, 3, 2) that we will beat at showdown and show a profit of P. Therefore the EV of checking is:
(4/12)(0) + (8/12)(P)
5. In question 4, is betting ever better than checking for Hero?
The simple way to do this is to set the EV of betting to be greater than the EV of checking and see if the equation still makes sense after we simplify things. So if the EV of betting is better than the EV of checking, then:
(4/12)(-1) + (2/12)(P+1) + (6/12)(P) > (4/12)(0) + (8/12)(P)
(4/12)(-1) + (2/12)(P+1) + (6/12)(P) > (8/12)(P)
-4 + 2(P+1) + 6P > 8P
-4 + 2(P+1) > 2P
-4 + 2P + 2 > 2P
-2 > 0
Since -2 is never greater than 0, we can determine that the EV of betting is never better than the EV of checking in question 4.
6. If Villain calls with an 8 or higher, what is the EV of betting a J for Hero? of checking a J for Hero?
This is the same process and everything as question 4. The EV of betting is (3/12)(-1) + (3/12)(1+P) + (6/12)(P) and the EV of checking is (3/12)(0) + (9/12)(P).
7. In question 6, compare betting and checking for Hero. What do you notice? Should Hero bet or check, and why?
Let's simplify the EV of betting:
(3/12)(-1) + (3/12)(1+P) + (6/12)(P)
(-3/12) + (3/12) + (3/12)(P) + (6/12)(P)
(9/12)(P)
Now let's simplify the EV of checking:
(3/12)(0) + (9/12)(P)
(9/12)(P)
What you should notice is that no matter what, they're always the same. That is, the EV of betting here is always the same as the EV of checking. Now would be a good time to think about why you should bet this hand anyway. (If you don't get it, go read this then come back.)
8. If the antes are 1 unit each and Villain calls with J+, how should Hero play his entire range?
For this question you need to find the EV of checking and the EV of betting with each individual hand Hero could have to see what the best play is for each hand based on the size of P. Here is an example:
EV of betting a K =
(1/12)(-1) + (2/12)(P+1) + (9/12)(P)
(1/12)(-1) + (2/12) + (2/12)(P) + (9/12)(P)
(11/12)(P) + (1/12)
EV of checking a K =
(1/12)(0) + (11/12)(P)
(11/12)(P)
In the above you see that betting a K is always better than checking a K, no matter what the pot size is. Similarly you'll find that you should always bet an A.
For playing a 7, you should find that betting is better than checking when P > 4/3.
For playing a 2, you should find that betting is better than checking when P > 1/2.
That should be a good enough guide for figuring it out. Notice that our range is always polarized. For example, what hands are +EV to bet P = 1.05? The range is {A, K, 6, 5, 4, 3, 2}. It would be worth looking at if Villain can call profitably with a T against this betting range. Note that if so, he can also call profitably with a 7.
8a. The EV of betting a T, 9, 8, 7, 6, 5, 4, 3 or 2 are all the same.
8b. The EV of checking decreases as you look at worse and worse hands.
|