Bram Cohen

Monte Carlo says you get a flush first about 54.5% of the time, a straight first about 36% of the time, and both at once about 9.5% of the time.

Here's some brain-dead Python code to do the calculation, for anyone who fancies checking my numbers. I don't know exactly how good the Python RNG is, but I'd be very surprised if it were bad enough to put the result in any doubt.

import random

cards = list(range(52)) # bottom 2 bits are suit

def play():
  # return 0 for a tie, 1 for flush first, -1 for straight first
  random.shuffle(cards)
  suits = [0,0,0,0]
  denominations = [0]*13
  for card in cards:
    suit = card&3
    denom = card>>2
    suits[suit] += 1
    f = (suits[suit]>=5)
    denominations[denom] = 1
    a,b = denom,denom
    while a>=0 and denominations[a]: a -= 1
    while b<13 and denominations[b]: b += 1
    s = (b-a >= 6)
    if f+s: return f-s

def count(n):
  r = {-1:0,0:0,1:0}
  for i in range(n): r[play()] += 1
  return r

One invocation of count(1000000) returned {0: 94637, 1: 544618, -1: 360745}. The standard deviation on the number of flush-first results should be about 500, so there doesn't seem much room for doubt that the actual frequency is more than 50%. Still less room for doubt that the frequency of straight-first deals is well below 50%.

You missed the case that a straight can be 10 J Q K A.

rng.shuffle() doesn't work on my system, it has to be random.shuffle() (which defaults to using random.random() for its seed). If you want to do a trial run with a different original seed, the place to get that is os.urandom(), by the way.

In a ten million sample run on my machine, which takes less than 20 minutes, I get {0: 996482, 1: 5104402, -1: 3899116}, which is almost exactly 51/39/10. My guess is that these numbers are correct and gorgonous's are off.

To get legalistic about the ace thing, either case is isomorphic, but A more closely resembles 1 than 14, and started its life out that way. Ace-high was added as an american root for the underdog kind of thing.

Ah.. I missed A2345 as well. Will correct and check again when I have access to the box that code is on.

list(range(52)) is redundant by the way. range() returns a list, and xrange() returns a generator. This will change in Python 3000, where range() will return a generator, and there will be no xrange().

Nice statistical trick.
Clearly, the flush/straight odds invert as you add more card, and the odds are only good for straights at low number of cards. Combining the probability for n=5 to 17 (no need to go further) end up being in favor of flush. Really nice hunch you had here. I wonder if somebody is going to bother coming up with the exact statistics on that!

Interesting problem. I wrote a program to do some simulations in C++ to look at the percentages when one limits the maximum number of cards that are dealt with the maximum number of cards ranging from 5 to 17. Straights seem to be more probable than flushes when the limit is less than 12 and flushes are more probable when the limit is 12 or higher. I ran 100 000 000 simulated hands and got these results:

Maximum number of cards dealt: 5
Straight: 392224 0.392224%
Flush: 196967 0.196967%
Both: 1497 0.001497%
Nothing: 99409312 99.4093%

Maximum number of cards dealt: 6
Straight: 1778824 1.77882%
Flush: 993525 0.993525%
Both: 20309 0.020309%
Nothing: 97207342 97.2073%

Maximum number of cards dealt: 7
Straight: 4667029 4.66703%
Flush: 2895686 2.89569%
Both: 109551 0.109551%
Nothing: 92327734 92.3277%

Maximum number of cards dealt: 8
Straight: 9208872 9.20887%
Flush: 6329483 6.32948%
Both: 370080 0.37008%
Nothing: 84091565 84.0916%

Maximum number of cards dealt: 9
Straight: 15110967 15.111%
Flush: 11495481 11.4955%
Both: 938188 0.938188%
Nothing: 72455364 72.4554%

Maximum number of cards dealt: 10
Straight: 21643452 21.6435%
Flush: 18219402 18.2194%
Both: 1922141 1.92214%
Nothing: 58215005 58.215%

Maximum number of cards dealt: 11
Straight: 27835585 27.8356%
Flush: 25922844 25.9228%
Both: 3337189 3.33719%
Nothing: 42904382 42.9044%

Maximum number of cards dealt: 12
Straight: 32843436 32.8434%
Flush: 33711283 33.7113%
Both: 5053111 5.05311%
Nothing: 28392170 28.3922%

Maximum number of cards dealt: 13
Straight: 36237134 36.2371%
Flush: 40599760 40.5998%
Both: 6800626 6.80063%
Nothing: 16362480 16.3625%

Maximum number of cards dealt: 14
Straight: 38084953 38.085%
Flush: 45824934 45.8249%
Both: 8283493 8.28349%
Nothing: 7806620 7.80662%

Maximum number of cards dealt: 15
Straight: 38824111 38.8241%
Flush: 49082287 49.0823%
Both: 9293807 9.29381%
Nothing: 2799795 2.7998%

Maximum number of cards dealt: 16
Straight: 38994214 38.9942%
Flush: 50614228 50.6142%
Both: 9805461 9.80546%
Nothing: 586097 0.586097%

Maximum number of cards dealt: 17
Straight: 38994214 38.9942%
Flush: 51047705 51.0477%
Both: 9958081 9.95808%
Nothing: 0 0%