An intersting question

[vkotlyar]

According to the EV stats form pokerroom.com, the avg EV for 10Jo and JQo are both -0.03. the EV of J10s is +0.15 and the EV for JQs is +0.23. I am assuming that the difference comes from the ability to make a bigger flush. For 10j and JQo, the difference is a bigger kicker for the J, and the bigger pair made by the queen. However, 10J can make 4 str8s, while JQ can only make 3. i was wondering if any1 can mathematically show how the ability to make an extra str8 will ofset having a higher pair and a bigger kicker. Aditionally, can some1 please calulate the probability of making a str8 w J10 and w JQ. i know that this is tedious, but i really cant do the math and this is the only way that i can try to analyze the game. On a side note, I am very happy with myself for starting to ask these questions as i see deep improvements in my game, and my transition from becoming a big loser to a big winner. thanks every1 for your replies
-vitaly

[switters]

okay -- i'll try not to botch the math this time...

first, you need to figure all of the ways you make a str8 with those two hands:

JT: 789, 89Q, 9QK, QKA
QJ: 89T, 9TK, TKA

for each of these hands, there are 4 * 4 * 4 ways to get the three straight cards that you need, and there are (47 choose 2) ways to complete the board (from the 47 remaining cards, choose any 2 of them...)

now, once again, we can't just multiply by four and be done - because we have double- counted many of the hands, and we need to subtract out the multiple countings.

we have double-counted, for example, boards like 789Qx, so we need to subtract out (4 * 4 * 4 * 4 * 46) hands for that...

so, for JT -

1) add up the straights (4 of them - 789xx, 89Qxx, 9QKxx, QKAxx)
4 * (4 * 4 * 4 * 47 * 46 / 2) = 276736
2) subtract the double-counted (3 - 789Qx, 89QKx, 9QKAx)
- 3 * (4 * 4 * 4 * 4 * 46) = 35328

total of 241408 boards that make you a straight, and the total number of boards is: (50 choose 5) = 2118760, which means you'll make a straight about 11.39% with JT

of those, 1.31% are flopped straights, (10.45% * 31.5% = 3.29%) of them are open-enders that complete, and the rest (6.79%) are gut-shots or back-door straights that get there...

for JQ -

1) add up the straights (3 of them - 89Txx, 9TKxx, TKAxx)
3 * (4 * 4 * 4 * 47 * 46 / 2) = 207552
2) subtract the double-counted (2 - 89TKx, 9TKAx)
- 2 * (4 * 4 * 4 * 4 * 46) = 23552

total of 182976 boards that make you a straight, and the total number of boards is: (50 choose 5) = 2118760, which means you'll make a straight about 8.68% with QJ

-switters

[vkotlyar]

In your other post, you calculated that you will flop an open ended str8 draw about 11% of the time. Does it make sense to say that you will complete a str8 w J10 ( and 109, 89, 78, 87, 56) about 11% of the time? i would think that the number is much lower, but you are the math whiz and im probably wrong. thank you very much for performing these tedious calculations for me. You have no idea how much i appreciate it.
vitaly [img]/forums/images/icons/cool.gif[/img]

[switters]

It does make sense, you can break it down like this:

1.31% of the time, you flop a straight.

10.45% of the time, you flop open-ended, and you will complete 31.5% of those (10.45% * 31.5% = 3.29%)

now, the part that seems counter-intuitive in this is that of the 11.39% of the time you MAKE a straight with JT, more than half of those times (6.79% / 11.39%) are from a gut-shot or a back door flush.

while a gut shot is less likely to complete (4 * 46 / C[47, 2] = 17.02%), you will flop a gut-shot (78, 79, 8Q, 9K, AQ, AK) almost twice as often as you'll flop an open-ender (some double-counting here)

of the ~20% of the time you flop a gut-shot, you make 17%, that accounts for 3.4% of the 6.79%.

so, it breaks down (approximately):

1.31% : flopped straight
3.29% : flop open-ended and complete
3.40% : flop gut-shot and complete
3.39% : back-door straight

don't be confused by these numbers -- they do NOT indicate that you are more likely to make a gut-shot or a back-door straight. You are LESS likely to complete any ONE gut-shot or back-door flush than you are to complete a given open-ender, but there are MANY more flops that give you a gutshot or a back-door draw, so if you saw every hand to the river, in the long run you would see more straights starting with a "worse" draw... these numbers don't take into account the MANY, MANY times you never see those gut-shots or back-door straights hit, because you are a good poker player who folds those bad draws [img]/forums/images/icons/wink.gif[/img] these are the percentages if you see the river, no matter what the odds.

-switters