Advanced Math, Poker Question?

Question

Playing Texas Hold'em, you are delt A2 suited Spades, lemping the flop comes 4, 5 of spades and a Q of harts;

1. What are the odds of getting the nut flush.

2. " " " " , of getting a straight

3. ", of getting A high paired on the turn

4. " of your opponent getting trips and paying you off on the river or busting your a on the river?

I would love to discuss with poker players but I need a math wizard to give me a formula! Is there such a thing.

P.S. the answer to number 1. is 50/50 i think. Two cards to come, the turn and the river each with a .25 chance of being a spade. Two time .25 is 50/50. but what are your odds of actually winning the hand?

Answer 1

1)

You have four of the five cards you need to a nut flush.  So there are nine spades out there than you have not seen.  9/47+9/46= 38%.  I think the actual math comes out closer to 35%, but it's in that ball park.  You're going to hit it about one in three times.

2)  To get a straight, you are going to need a five.  There are four fives.  So it's 4/47+4/46= 17%.

3) There are two unseen aces.  So the odds of getting one on the turn is 2/47 = 4.26%.

4) I don't think you can really answer that because we don't know what cards your opponent has.  Does he have a pair?  In which case he just needs to match it to make trips.  Did he flop a pair, in which case he needs another card the same to make trips?  Do any of his cards clash with what you have (ie does he have some of your outs or vice versa)?

PS) No the odds are not 50/50 of getting a nut flush.  There are only nine cards that can help you versus 38 cards in the deck that don't make your flush.

As for your odd of actually winning the hand, it really depends on what your opponent has.  If he flopped trips for example, and the board pairs he could make a full house or even quads both of which beat your nut flush.  Conversely if you hit the three of spades, you now have a straight flush, that would beat anything he could be holding, unless he has the 36 of spades, thus blocking your straight flush possibility, and giving him the option of hitting the seven of spades.  He could also have 67 of spades, and catch and 8 or 3 of spades to make a straight flush.

Your odds of being beaten depend on what you ultimately hit (because on the flop all you have is a draw), and what happens on the turn and river.

Answer 2

I dunno why you think this is advanced math. the formula is outs/remaining cards. 4 outs with 40 remaining cards = 10% chance, with the turn and river it's 2X this, or 20% chance.