An amusing string of hands

SpaceAce · #1 12-10-2004, 06:00 AM

I know the answer to this question is simple math but I am hoping someone will work it out here in this thread because I think the mechanics of the problem are interesting, especially when you factor in some of the finer details such as additional draws I flopped and what my opponents were holding. This is an actual series of four hands I played a couple of weeks ago.

The summary: In four hands, I was dealt Ah Kh three times. Every single time, I flopped a flush draw (sometimes with a little something extra) and two overcards and every single hand ended with me having no pair and no flush.

The specifics:
Hand #1: The flop was Th Jh X? giving me a gutshot Royal draw to go with everything else.
Hand #2: The flop was 3h 5h X? giving me vague wheel and steel wheel draws in addition to the flush draw and overcards.
Hand #3: This is the hand I did not get Ah Kh.
Hand #4: The flop was Xh Yh Zs. This hand was won by another player with As Ks who made his backdoor flush.

So, what are the chances of getting that specific hand three out of four times, flopping a flush draw each time and never so much as pairing up by the river? What if you add in the other draws? This problem might be too straightforward to actually interest anyone but I think it's pretty cool, especially when you factor in the big draws I flopped every time and the weird outcome of the last hand.

SpaceAce

mannika · #2 12-10-2004, 09:09 AM

Probability of getting AhKh on any given hand = 0.0754%
Probability of getting it exactly three times in four hands = (0.00754)^3*(0.99246)*4 = 0.000001701, or 1 in 587642

Probability of flopping flush draw on any given AhKh hand = (11/50)*(10/49)*(40/48)*3 = 0.1122

Probability of flopping flush draw and NOT hitting flush or pair = 0.1122*(32/47)*(31/46) = 0.05148

Total probability of your situation happening:
= 0.000001701*(0.05148)^3
= something really small = 1 in 4,306,915,090

In conclusion, either:
a) my math is off
b) you are a liar
c) this happened, and it was a ridiculous fluke
d) any combination of the above

SpaceAce · #3 12-10-2004, 09:37 AM

[ QUOTE ]

a) my math is off
b) you are a liar
c) this happened, and it was a ridiculous fluke
d) any combination of the above

[/ QUOTE ]

The chances that I just dropped in here after 1,400+ posts in these fora to make this up are even less than 1:4,306,915,090.

SpaceAce

Cornell Fiji · #4 12-10-2004, 11:06 AM

Party Poker is rigged!

(someone had to say it)

SpaceAce · #5 12-10-2004, 11:10 AM

[ QUOTE ]
Party Poker is rigged!

(someone had to say it)

[/ QUOTE ]

Heh, didn't even occur to me because this happened live [img]/images/graemlins/smile.gif[/img]

SpaceAce

gaming_mouse · #6 12-10-2004, 11:34 AM

mannika,

Looks like your math is correct, but none of your explanations is the correct one, IMO.

The crucial thing to note here is that SpaceAce would have been equally surprised -- and likely asked the same question -- had this chain of 4 hands happened at any time during his entire poker career.

Just by his number of posts, I'm putting his lifetime number of hands at over 100K, possibly over 1 million. So the real question to ask is: "What is the chance that 3 hands meeting his requirements (AKs, flop flush draw, etc) occur in a string of 4 hands at some point in a string of X hands?" where X is the total number of hands he's ever played.

The situation is very much the same as when people are surpised by unbelievable coincidences in their lives, and ask, "My god! What are the odds of that?" No matter what the coincidence, the thing to note is that there are an uncountable number of things that could happen to you at any moment in your life that would surprise you just as much. You are, so to speak, playing millions of unseen hands of "the coincidence game" as you move through your daily life. But of course you only notice and remark upon the royal flushes.

gm

mannika · #7 12-10-2004, 02:04 PM

I agree that the probability of anything in life happening, when you look into it minutely enough, is ridiculously small. However, given the situation, I think it is extremely unlikely that this would happen. Even if he plays B&M professionally, and has done so for the last 15 years, he would probably have just over 1 million hands. Keep in mind that the probability that I gave was 1 in 4 BILLION. Therefore, this string of cards/outcomes would happen to only 1 in 4,000 pros with 15 years of full-time play under their belt.

This, combined with the fact that these hands are all the same, led me to originally comment that this was indeed quite a fluke, for it to only happen to 1 in 4,000 pros over a decent poker career.

Keep in mind that I know exactly what you are referring to with your example of things that happen in life. However, irrespective of the probability of something happening, it's much more interesting to hear of someone winning the lottery twice or getting struck by lightning twice, rather than, "Wow, it's amazing that he died at the age of 65, 3 days, 15 hours, 12 minutes and 5 seconds. That's like a 1 in a million chance!"

In this case, the situation was noticeable enough to be considered a fluke, although by all standards, having AhKh, then 7c2d, then 5s5h, then TdQh, (or something unnoticable) has as small of a probability of happening.

gaming_mouse · #8 12-10-2004, 04:53 PM

[ QUOTE ]
Keep in mind that the probability that I gave was 1 in 4 BILLION. Therefore, this string of cards/outcomes would happen to only 1 in 4,000 pros with 15 years of full-time play under their belt.

[/ QUOTE ]

You make a good point here. First, though, your math is wrong. Is it should be (approximately, and you have to use google calculator since the numbers are so big):

1 - (((4 306 915 090 - 1) / 4 306 915 090)^1 000 000) = 0.000232157765

but that only helps your argument (more like 1 in 5000). HOWEVER, to my original point, he would have been equally surprised had this happened with AQs, TJs, etc. So my argument still applies, I think.

Please note I'm not really taking issue with the semantics of calling things like this a fluke. I think it's fine, and I probably would too. But other posters (part jokingly, granted) were talking about this being evidence of rigged poker. My argument was mainly intended to show that it does not offer any such evidence.

gm

SpaceAce · #9 12-11-2004, 10:07 AM

[ QUOTE ]
Even if he plays B&M professionally, and has done so for the last 15 years, he would probably have just over 1 million hands.

[/ QUOTE ]

Actually, even if you include my internet play, I estimate that I am well short of 1,000,000 hands. Also, these hands came near the beginning of a session that probably didn't break 400 hands.

SpaceAce

SpaceAce · #10 12-11-2004, 10:14 AM

You made good points in both of your replies. You are correct that I would have found this string of hands noteworthy if it had been almost any suited connectors. As to the joking comment about rigged poker, these hands took place in a brick and mortar casino so I am going to assume there was no foul play [img]/images/graemlins/smile.gif[/img]

SpaceAce