You hold:
3 of spades
3 of diamonds
4 of diamonds
5 of diamonds
6 of diamonds
Should you draw one or three? Does tournament circumstance matter here? It seems that if you have, say, five opponents, your pair of threes won’t survive anyway. And if you’re near the final hands before advancing to the next round in a close contest, shouldn’t you take these kinds of risks?
If you draw only one card, the best you could hope for is a 2D, which is of course a 1/52 chance. You have a somewhat better chance of getting a flush, as there are 9 diamonds left, some of which may be in the deck still. If it was me, I would probably draw three.
Actually, any old 2 would get me a straight. Any diamond would get me a flush. A 2D would likely get the me the house’s champagne! 
Depends on the circumstances. What do the other players rate to hold? What are they drawing?
For instance, here’s a good question:
You hold 3,3,3,T,J. How many cards should you draw?
Often, the answer is one, or none. Your trip threes are almost certainly the best hand, and the odds of hitting a fourth three or picking up a pat pair are small. So you aren’t going to improve the hand anyway, and you likely don’t need to. So deception becomes the most important factor. What are you trying to represent? Drawing one card often represents someone trying to make a straight or a flush, which means that they’ll usually miss and have nothing. Drawing two cards can indicate anything from nothing but a 3-flush to a pair with a big kicker.
I don’t like 5-card stud much, because it’s too much of a crapshoot. 7-card is a better game because there are more betting rounds where you get progressively more information.
Wouldn’t the seven of diamonds do the trick as well, making the odds 1/26?
typo’s right about the 7. Overall, I see:
2 cards that give you a straight flush (2D, 7D)
7 cards that give you a flush (any other diamond)
6 cards that give you a straight (the non-diamond twos and sevens)
For a total of 15 out of 47 that give you an excellent chance of winning. Just under 1 in 3.
I figure drawing to your pair only gives you a good chance at winning if you draw another 3. This is only 2 out of 47. There are other chances, like drawing another pair, or conceivably 3 of a kind (to complete a full house) but no combination is going to give you the neary 1-in-3 odds of keeping your potential straight flush.
hmm, last line should read: “…the nearly 1-in-3 odds as keeping your potential straight flush.”
There is no issue here. You draw one. You have a 36% of improving your hand to a straight or a flush (not including a straight flush). The odds of improving to a WEAKER hand (trips) is significantly less (around 12%) and adding in the possiblity of drawing to a hand stronger than a straight or flush (full house or four of a kind) doesn’t help much.
I’m not sure where you got 36% (my 15 out of 47 gives a little less than 32%) but the rest of it is sound. I neglected to mention the remote odds of drawing both threes to make four-of-a-kind, but in any event, drawing 1 is hugely better then drawing 3.
I guess you should also figure in the fact that you could draw a card to match one of the other four, thereby improving the rank of your pair. So, would that increase the odds of bettering your hand by drawing 1 card to 19 out of 47, which is slightly more than 40%?