Probability of Losing 15+ hands in a Row 5 Within One Week

First off, forgive the typo in the title of this thread.

EDIT: MY WORK IS WRONG. SEE ASSUME_R's POST BELOW FOR THE CORRECT ANSWER TO THE PROBLEM.

As a quick introduction, a member of this site claimed to have lost 15+ hands in a row on 5 occasions within the period of a week on the voodoo section of this site. Zg asked what the odds of this happening being assuming that one had 40 hours of play in the said week.

To investigate this without getting too deep into it, I am recycling a 1 Billion hand simulation that I performed earlier in the day. My time is becoming more scarce, as I'm invested in multiple intellectual projects at the moment. The parameters were nothing special. Hi-Lo, play-all, 6D, S17, I18, NS. I don’t think that there are any more parameters that are worth mentioning.

The simulation indicated that the player lost 15+ hands in a row exactly 32,461 times in the 1 Billion hand simulation. For these calculations, I will assume that 100 rounds = 1 hour. Therefore, we can say that the player lost 15+ hands in a row 32,461 times in 10,000,000 hours.

We can also say that one could expect to lose 15+ hands in a row once every 308 hours of play.

(10,000,000 hours / 32,461 occurrences = ~308 hours between each occurrence, yes?)

We could also say that we have a 0.3% chance of losing 15+ hands in a row in any given hour.

(32,641 occurrences / 10,000,000 hours = ~.00326 occurrences per hour)

Now, what I’d like to do is find out the probability of losing 15+ hands on five occasions within 40 hours.

The probability of losing 15+ hands in a row within a 40 hour period must be 40 times greater than losing 15+ hands in a row during a one-hour period, yes?

Then, the probability of losing 15+ hands in a row within a 40 hour period would be 13%.

(40 x .326% = 13.04%)

Therefore, the probability of losing 15+ hands in a row 5 times within the 40 hour period should be 0.004%, right?

((.1304)^5) x 100 = 0.0038%

I’m most certain that my math is correct, but if someone very knowledgeable in the field of statistics would like to double check my math, I’d be forever grateful.

In fact, I originally had posted the following (red font), which was pointed out to be incorrect by my math-genius younger brother (14 years old).


((.00326)^5)(40) x 100 = Wrong Answer

My brother pointed out that this would be the probability of losing 15+ hands in a row 5 times within an hour, but giving you 40 tries to do so. Of course the probability of this is much lower than what we were actually looking for.

If he gets into this business, he will surely blow me out of the water.

Disclaimer: Yes, I know ... 1 Billion rounds may not be enough rounds to get significant answers. This was my quick & dirty answer to Zg's question recycling, per se, a simulation that I’d run earlier.

Best,

SP

Southpaw,

I have really begun to enjoy reading your studies and I very much like your methods! I was going to PM you that, but why the hell not say it publicly.

However, unfortunately this time you are not correct in some of your math.

Here's why:


Correct.

Here's where the trouble starts. What if it was a 1000 hour period? Would it be 326% chance of losing 15+ hands in a row (1000 * .326%)? No, because P <= 100% always. Here's what you do:

Probability of not losing 15+ hands in a row for hour 1 AND
Probability of not losing 15+ hands in a row for hour 2 AND
...

So it's (100% - .326%)^40 = (.99674)^40 = 0.8776, which is the probability of not losing 15 hands in a row for all 40 hours.

So the probability of losing 15+ hands in a row for any of the 40 hours is 1 - 0.8776 = .1224 = 12.24%. Not too different of a result in this case, but the method is key!

Check out this http://en.wikipedia.org/wiki/Birthday_problem to help explain somewhat.

Okay, let us continue.

So now, we have 0.1224^5 * 100% = .0028%

That is the probability of losing 15+ hands in a row 5 times in a given 40-hour period.

It didn't make too much of a difference in this problem, but pay careful attention that no amount of hours can give you P > 100%.

So if, instead of 40 hours, we had used 80, here's what would have happened:
Your result = (.00326 * 80)^5 * 100% = .12%
My result = (1 - (1 - .00326)^80)^5 * 100% = .06%

Disclaimer: feel free to correct my math if anything seems off :grin:

Now, one final note. If you had said, "What is the probability of losing 15+ times in a row in 5 40-hour periods, when I only played 20 40-hour periods this year", you would use the binomial distribution to model that.

As an added comment to my brother's intellectual capability, I once made an interesting wager with him.

He bet me $100 that he could solve a rubik's (that I scrambled very thorouhly. He also was not in the room while I scrambled it.) cube blindfolded after being able to just look at it for 5 minutes (he was not allowed to start solving or make any moves).

I lost this bet ... talk about negative negative EV :cry:

He now holds a respectable nationwide record for being able to do this much more quickly than he was required to in our wager.

SP

Now that's what I call good genes!

But ask him to solve a 5x5x5 cube instead of the traditional 3x3x3 The 3x3x3 will seem trivial in comparison.

THANK YOU, ASSUME_R. This makes complete sense. My brother had actually briefly tried to explain to me that this was the way it needed to be done, though I didn't fully understand. (He went to bed right after telling me my first method was incorrect lol. Talk about a one-liner.) Again, thanks for the correction.

SP

He has every cube up to a 7x7 and is able to solve all of them without trouble. He also has this one with 12 sides, though that one is supposed to be the hardest.

He's also a highly competitive chess player, though I probably shouldn't say more or I'll end up giving away my anonymity ...

SP

So thats like 1 in 350, right? Thats nothing, I thought it would be much-much higher. zg

I definitely agree with you. I was horrified when I found out that one loses 15+ hands in a row every 300 or so hours. No wonder casinos welcome progressionists that rarely ever lose. (Their losses, however, are huge). I figured that this was much more improbable. They really will get those progressionists eventually ... Good for the casinos (Patting my local floorman on the back).

SP

Actually, it was 1/35000. But now that I think about it, you shouldn't just do 0.12^5. You should use a binomial distribution, where you want the probability of something happening at least 5 times in 40 independent hours, where p = .00326 is the chance of it occurring in any one given hour. The result is actually even smaller: 1/4539266 (1 in 4.5 million).

I am not surprised at all that it is so small, since in a billion rounds, 15 losses in a row only occurred 32 thousand times.

This assumes each hour is independent, and you have a 1 in 306 chance of losing 15 hands in a row in that hour. Also we are not taking into account 15-hand losses that happen to stretch between hours. We are also considering an hour with 2 15-hand losses as negligible, since it will almost surely never happen.

Also, as an aside, the probability of seeing no 15-hand losses in any of the 40 hours is 88%. The probability of having exactly 1 hour with a 15-hand loss is 11%, and the probability of seeing exactly 2 hours with a 15-hand losses is 1 in 137. After that, it decreases significantly:

Probability of seeing exactly N hours with a 15-hand loss in a 40-hour period:
Exactly 0 of the 40 Hours: 1 in 1.14 (88%)
Exactly 1 of the 40 Hours: 1 in 9 (11%)
Exactly 2 of the 40 Hours: 1 in 137 (.7%)
Exactly 3 of the 40 Hours: 1 in 3,297
Exactly 4 of the 40 Hours: 1 in 108,963
Exactly 5 of the 40 Hours: 1 in 4,627,163
Exactly 6 of the 40 Hours: 1 in 242,528,213
...
At least 0 of the 40 Hours: 1 in 1 (100%)
At least 1 of the 40 Hours: 1 in 8 (12.24%) <-- this is the answer from my original post! yessss
At least 2 of the 40 Hours: 1 in 131 (.8%)
At least 3 of the 40 Hours: 1 in 3,197
At least 4 of the 40 Hours: 1 in 106,410
At least 5 of the 40 Hours: 1 in 4,539,166
At least 6 of the 40 Hours: 1 in 238,684,693

In excel, just type the following:
would give you the probability for exactly 3 of the 40 hours.

We could also say that we have a 0.3% chance of losing 15+ hands in a row in any given hour.

(32,641 occurrences / 10,000,000 hours = ~.00326 occurrences per hour)

This is actually wrong. You can't divide the number of occurences by the number of hours. They're not measured under the same scale. One is a measure of trials. The other is a measure of time. By your logic, one occurance per hour (10,000,000 occurances/1,000,000,000 hands) is a 100% chance of occuring in an hour. By the same logic, if the number of occurances per hour is greater than 10,000,000 the probability of occurance is now >1.0, and that's not possible.

Yes, you are correct. This is precisely what AssumeR went on to point out.

I had tried to indicate this--that is, that my work was flawed-- with the edit at the top of my original post. I figured I should leave the post as was otherwise, though, so that others could see where I went astray.

Spaw

As a fellow Cuber that is impressive. Doing it blindfolded, not the time. I can do it mostly blind, but need to look once or twice during the process. Back when I was cubing competitively I could consistently solve under 50 seconds and occasionally under 30. These were impressive times for back then, but these days there are lots of kids that would blow those times out of the water.

Last night I had a rare chance playing 6deck shoe game one on one with the dealer. The TC was greater than -2. I lost 15+ hands in a row. Most of them involved both me and the dealer having 17 - 21, but somehow she kept having higher hands.

All I can say is the variance in this game can really make you sick.

Hi people.

I would like to ask to SP:

Which simulator did you use to obatain this?

He probably did not play his hands the best way to win to have lost 15 hands in a row consectively 5 times in a week. That said my worst losing streak in a row was 9 hands.

Norm's very own CVData.

Spaw

Statistics can confuse even the smartest people

This illustrates why I have such respect for those gifted in statistical logic. I am very gifted in math ability but a little rusty in my older years. I was top in my class in statistics but very often didnt use the right logic to attack a problem as in this case.

Being top in the class was great relatively speaking but I dont compare myself to other people. I compare myself to perfection or the best I can possibly do. I always felt I was lacking as I measured myself using my criterion when it came to statistics regardless of how I looked compared to the my surroundings.

Kudos to the true experts in statistics who are willing to help those who are only good at it. We make more mistakes than most of us realize.

or happy

One shouldn't speculate on matters that one has no knowledge about. I don't believe I made any errors or mistakes.

I'm rather impressed (actually envious) that your biggest losing streak is only 9 hands in a row. I seldom have a night go by where I don't have at least 1 such streak. I won't be using martingale anytime soon.