Posted by Dinker on Tuesday, 16 December 1997, at 1:04 p.m.
My friends and I play poker every Tuesday night, and I am looking to employ a new tactic tonight. Our favorite game is five card draw, Jacks or better to open, and trips to win. Depending on what is dealt of course, everyone generally aims at trips. My question, to all the experts out there is, if I am dealt a pair of sevens and some combination of three spades, hearts, diamonds or clubs....do I not have better odds at landing a flush than I do by holding my pair of sevens. That is, in a game where I need trips to win(or any game of five card draw) isn't it statistically more likely to get two spades in two cards than it is to get a seven with three cards?
As long as I am on the subject, wouldn't the same hold true for straights? If I am dealt, say, a pair of threes and a 7,8,and 9...aren't my odds greater of successfully getting the straight than I am of landing my trips? What do the experts say?
As for those of you reading this message, how do you usually play? Is there any site on the internet that addresses questions like this? It just seems to me that if everyone else is shooting for trips, my winnings will increase greatly if I shoot for flushes and/or straights nearly everytime! Even if I am dealt a pair of threes and a 7,8, and 10...it still seems like I have better odds at getting my straight(by dropping the threes), than I do at drawing another three. And the odds of getting a flush by holding three of the same suit seem even greater than those for a straight!!
Are there any number crunchers out there that can help me out or at least offer an opinion? Thanks in advance!
Greg's advice on how to play this game as well as the generic home game(tight, but not to the point of being resented) are right on the mark. As to your specific question, most holdem players could rattle off that the odds to flop a set with a pair are about 7.5:1, and the odds of hitting runner, runner to a flush are about 4%. Therefore, the hitting trips is a definite favorite.
However, this odds calculation is not impossible to compute and is a good exercise for aspiring poker players. One of them is an "AND" calculation; the other an "OR." After seeing how these are done, you should be able to figure out the answer to all your future odds questions yourself.
The "AND" goes "I need a suited card AND another suited card."
The odds are:
(10 of your suit remaining/47 other cards) // ya gotta get that first suiter.
TIMES
(9 more of your suit/ 46 more cards) // ya gotta get that other one, too.
EQUALS
(90 / (46*47)) = .0416 or about 4% .
The "OR" goes "I need a seven or a seven or a seven."
The odds are:
(2 left in the deck/ 47 other cards) // one way is to get that 7 on the first card
PLUS
(still 2 left / now only 46 other cards) // or maybe I can get it now
PLUS
(still 2 left / now down to 45) // last chance
EQUALS
(2/47)+(2/46)+(2/45) = .1304 or 13%
This is slightly better than the holdem-flop-a-set odds because you already have eliminated three non-sevens from the deck (your discards).
This number is not completely accurate, however. You must take into account the times you hit a 7 early and hit another 7 late, which this model credits as two successes but is in fact just one. The difference is less than 1% (.002775, I believe). Given the above examples, the exercise is left to the reader. (I hate it when they say that!) For few-outers you can usually neglect this factor, but for, say, flush draws with two to come, you need to do the eliminate-the-double-success subtraction. (That is, calculate the probability that this occurs and subtract it from .139)
To base your drawing decision solely on this statistical analysis assumes that when you make a hand, you win. In the real world, this is not the case. For example, if you are last to draw and one of your opponents stands pat, and you know he doesn't play tricks, go for the flush. This can be taken another level, however. If you know your opponents will reason thus, and you are the first to act with a set of deuces, you might want to consider standing pat to throw them off of drawing to better trips. (This needs to be weighed against the odds of going for the boat, and them not being aware, etc..) If you know your opponents might think like this and you are last to act and... well, that's poker.
BUT WAIT THERE'S MORE!
Tom Weideman followed my reply with:
Umm, there's an easier way. The correct probability of getting another 7 is found by calculating the probability of not getting it (using "AND"), and then subtracting that probability from one:
miss AND miss AND miss = (45/47)*(44/46)*(43/45) = 0.875
Then NOT missing three times is 1-0.875 = 0.125 (12.5%)
This is the probability of getting AT LEAST one more 7. The probability of improving to a "trips or better" hand also includes drawing three cards of the same denomination, which tacks on some more percentage. Assuming you did not throw away a second pair, this probability is also easily found:
6 ways to get trips of the 3 denominations you threw away 24 ways to get trips of the 9 denominations unseen
Total number of ways to get three equal cards on draw:
6*3 + 24*9 = 267 (He means 234; change other numbers accordingly -- JG)
Total number of ways to get any three cards on draw:
47*46*45 = 97,290
Probability added accounting for drawing three equal cards:
267/97,290 = 0.3%
The total probability for improving a pair to trips or better is 12.76%. Throughout I assume no bug in the deck, btw. Hmmm, maybe I've gone too far. I'll stop now.
Tom Weideman
I concur.
Last Modified 2/9/00