Four-flush, four straight....

[oddsock]

Could some one tell me what the probability of being dealt a four-flush or a four straight (consecutive cards, not inside straight drawing hand) in any five cards from a freshly shuffled new deck?

Thanks.

[yellowbastard]

A four flush has 13 cards to improve out of 52. In 4 tries it's --> (13/52)*(12/51)*(11/50)*(10/49) ~ 0.26% Not sure about 5 though.

[oddsock]

Thanks for the reply. I've managed to work it out (hopefully it's correct - need some checking) it's quite a brain masher one.
Thanks again.

[Drew Smith]

The outcome of the straight is dependent on your cards. JT has more straight possibilities than AK or A2. It also depends on how far the cards are gapped (T6 vs. T9) because with a large gap there are less straight combos.

[Copernicus]

[ QUOTE ]
Could some one tell me what the probability of being dealt a four-flush or a four straight (consecutive cards, not inside straight drawing hand) in any five cards from a freshly shuffled new deck?

Thanks.

[/ QUOTE ]

Is your question the odds of improving the four flush/straight thats already dealt (the answers) or of being dealt the four flush/straight to begin with (the way the question reads)

[oddsock]

The question was the way the question reads Copernicus. I wanted to know the the odds of being dealt a 4-flush or a 4 straight. I should however add 2345 to 9TJQK only, those 4-straights which could be drawn to 8 outs.

The mathematics stumped me for the straights I'm afraid ended up using an approximation.

[AaronBrown]

There are 2,598,960 five card poker hands, C(52,5). 111,540 of them are four flushes, 4 suits times C(13,4) ways to choose four cards from one suit times 39 fifth cards. This does not count flushes as four flushes. There are 5,148 flushes (40 of which are straight flushes).

There are nine different open-ended straight draws, headed by cards from five to King. Each of them can be suited in 4^4 = 256 different ways (this includes suited four straights). But now it gets tricky.

If the fifth card is one of the seven ranks (28 cards) that does not either complete the straight or pair with one of its cards, it's simple. 9*256*28 = 64,512 four straights of this type.

If the fifth card is one of the 2 ranks (8 cards) that completes the straight, we don't have a four straight, so we can forget about it.

If the fifth card is one of the four ranks (12 cards) that pairs a straight card, we can't just multiply 9*256*12 because we'll double count. We can multiply 9*256*6, or we get the same answer by considering there are 36 different four straights with a pair (the 9 four straights, each of which can have one of the four cards paired). Each of these can be suited in 4*4*4*6 = 384 different ways (4 ways for each of the unpaired cards, 6 ways to select two different suits from one rank). 36*384 = 13,824. There is no fifth card, of course.

Adding the two types together 78,336. In addition, there are 10,240 straights, 40 of which are straight flushes.

You may wonder that it's harder to get a four straight than a four flush, and harder to complete a straight (8 outs) than a flush (9 outs); yet flushes are rarer than straights. The answer is that not all straights come from four straights, but all flushes come from four flushes. In other words, if you take any card away from a flush you have a four flush, but if you take any card away from a straight, only 40% of the time do you get an open-ended straight draw.

However, it is important to know that if people draw only to four flushes or open-ended straight draws; you'll see more flushes than straights at showdown.

[oddsock]

This is absolutely fantastic information AaronBrown. The flush count I was OK with, the straights sort of busted my brain a little.

(OK, it busted it alot)

Thanks again, i can return the favour with any questions on Human Geography or Monty Python, both of which have little relevance in Five Card Draw it seems.

Thanks again.

[AaronBrown]

Okay, is Dorset Blue Vinney (Cheese Shop sketch) a real cheese and where can I buy it?

[oddsock]

Well, would you believe it - I just happened to have this page open underneath a PL draw game :

http://www.houseofcheese.co.uk/acata...Cheeses_9.html