Last night, in my ongoing effort to lead my children into a life of debauchery and shiftlessness, I started teaching them poker. I quickly ran into the limits of my knowledge of the game (five card stud) when I was dealt the 6, 7 and 8 of diamonds, the 10 of hearts and 10 of spades. I had no idea what to do, but I told them that the teeming millions would help us get back on our wayward path.
So what should I have done – gone for the flush, straight or tens?
If you keep 6,7,8 and 10, hoping for a straight, your chances of getting it are 4/47, since four of the 47 cards you don’t already have are the nines you need.
Hoping for flush: You need two more diamonds. You already have five cards, so 47 are still “active,” ten of which are diamonds (there are 13 diamonds in the game, three of which you already have). So chances to get two more diamonds are (10/47)² (or at least roughly - after drawing the first card, the probability for the second card to draw should change oncer again), which is lower than 4/47.
Keeping tens: There are four tens in the pack, two of which you already have. Chance of getting a three-of-a-kind: 2/47, chance of getting a four-of-a-kind: (2/47)². Again, roughly - I’m not a mathematician, and I’m too lazy to figure out how the first card drawn affects the probability of the next cards to draw, and there’s certainly someone here who can do this much better than I can, but it should do as an approximation.
Sure, a four-of-a-kind is worth much more than three-of-a-kind, so you could multiply each combination’s probability with its value to get expected value for each choice, but I don’t know how to assign a numerical value to each combination.
Summary: I’d go for the straight. But then again, I tend to lose at poker.
You don’t get to change your cards in five card stud.
There’s no drawing in stud. You mean five card draw.
One of the first things most kids are taught about poker is that you shouldn’t draw to an inside straight. This is generally good advice, since you only have four “outs” - there are only four cards in the deck that can make your hand. In your case you would need to draw one of the four nines. Since there are 47 cards you haven’t seen, your odds of making your straight are 4/47. You shouldn’t call with this hand unless you stand to win about 12 times more than you have to put in. This would be very rare in a game with only two betting rounds.
An open-end straight draw is much better. An example would be if you had 7, 8, 9, 10. Then there are eight cards that can make your hand - the four sixes and the four jacks.
I also don’t want to get into the mathematics of drawing two to the flush. I’d hold the tens.
I assume that you really meant five card draw. A lot of the answer depends on your relative position and how others before you have acted.
There are no circumstances under which you would dump the two 10s and go for the flush. The odds are way too long.
The famous advice to never draw to an inside straight is usually, but not always, correct. If the pot is huge and you think the pair is beat, you should go for it. See, here is where position matters. If someone acting before you is a good player who rarely bluffs and draws two cards you can put him on trips. Go for the straight with a very large pot.
For the most part you should keep the 10s and draw three.
Haj
Actually, maybe I do know how to figure the flush draw. You’ve seen three diamonds out of ten and you must draw two consecutively. It would be 10/47 * 9/46. This works out to roughly 1/24. Not good odds at all.
The 6, 7, 8 also gives you some straight and straight flush draws, but I really don’t want to get into those.
Your odds of getting one more ten are 2/47 + 2/46 + 2/45. This works out to better than 1/8.
Er, I mean you’ve seen three diamonds out of thirteen, leaving ten in the deck.
Depends on how many players are playing , how they’ve been playing and your relative position. Also depends on the pot and how many cards those before you have drawn. And it also depends on alot more than that. That chance of improving your hand if you keep the 6,7,8 is around 6%, half that of dumping the three straight and going for trips. Definately drop the gut-shot draw because the odds aren’t much better than a chance for a much better hand by going the first route.
If you draw 2 cards against amateurs they will assume you have trips. It would make me wonder if they weren’t afraid to bet into me having drawn only two cards because they’ll probably have higher trips than you at which point you will have needed the longer shot of the open-ended draw/flush.
The answer is: It depends. If you put a gun to my head I’d say dump the 10’s against good players and the flush/straight draw against newbs.
On a purely statistical level, I’m interested in finding out if the combined possibility of getting a straight, flush or three of a kind after dumping the tens actually outweighs the probability of getting ‘trips’ after keeping the tens.
I’m going to make a program to check everything if I can do it quickly enough. Will report back… My probability isn’t really up to it.
Hmm… Forget that idea. Making a program that recognizes the hands is too complicated for my liking right now. Let’s give the math a shot:
For dumping the 10’s:
P(Flush) = 10/47 * 9/46 = 45/1081
P(Straight 1) = 4/47 * 2/46 = 4/1081
P(Straight 2) = 4/47 * 4/46 = 8/1081
P(Straight 3) = 4/47 * 4/46 = 8/1081
P(Trips 1) = 2/47 * 1/47 = 1/1081
P(Trips 2) = 2/47 * 1/47 = 1/1081
P(Trips 3) = 2/47 * 1/47 = 1/1081
P(GoodHand) = 68/1081
Subtract (3/1081) for the Straight Flush combinations that we’ve counted twice.
For dumping the Diamonds:
P(Trips 1) = 2/47 * 3 = 6/47
P(Fourkind) = (1/47 * 2/46) * 2 = 2/1081
P(GoodHand) = 140/1081
For dropping One of the Tens:
P(Straight) = 4/47
P(GoodHand) = 92/1081
Okay, ignoring the inevitable mistake in my Math, the best course of action is dumping the diamonds, followed by dumping one ten, followed by dumping both tens.
Okay Mistake #1, for dumping the 10’s it should read
P(Trips 1) = 3/47 * 2/46 = 1/1081
P(Trips 2) = 3/47 * 2/46
P(Trips 3) = 3/47 * 2/46
After subtracting (3/1081) This brings P(GoodHand) to 71/1081. No change in order.
Before you decide what to do, you should consider that, A) the 10s may be the best hand (not unlikely in a short-handed five card draw game), and if it’s not, that 2 pair would make the best hand (pretty likely).
So by keeping the 10’s, you can improve your hand by hitting another ten, or by picking up another pair, or three of a kind for a full house.
It’s a common flaw to always want to go for the best hand possible, but it’s not always smart. For example, what would you do in 5-card draw if you drew a pat full house? Stand on it?
The answer is ‘it depends’, but it’s often correct in this case to throw away one or two cards in your full house, and keep your trips. They’re very likely to be the best hand at the end anyway, and the deception you gain will pay off with more people calling you.
If you keep the two tens, against good players you should probably also keep your highest kicker. Discarding three cards gives your hand away.