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# Casino Game

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Suppose that you are at a casino playing roulette. The strategy you are using is to, before each bet, flip a coin to determine whether to place your bet on red or on black (which, according to the rules of the game, should each have almost a 50% chance of occurring). After you've placed each bet, the roulette wheel is then spun. Suppose that you lose 59 times in a row (i.e. for 59 consecutive plays, when you place your bet 3 on black the ball then lands on red, and when you place your bet on red the ball then lands on black).

From this experience, it is most rational to conclude that:

a) Using a coin toss to determine whether to bet on red or black is in general a very bad strategy for playing roulette

b) The game is somehow rigged against you and the casino or its employees are cheating you

c) You are very likely to win on your next bet if you continue this coin flip based strategy

d) The roulette game is broken, but there is no reason to assume that it was broken intentionally

e) You were merely very unlucky

f) One cannot reasonably conclude which of the above options is more likely

Dec 14, 2020

#1
+1

There's a bit of opinion that goes into this question, and here is mine:

a)  It's neither a bad strategy nor a good strategy, it's indifferent.  You have a 50-50 chance of being right whether you flip a coin or not.

b)  This is unlikely in a regular casino.  They don't need to cheat.

c)  No. Your odds on the next try are 50-50 just like every other time.

d)  No.  As long as the wheel turns, it isn't broken.

f)   Wrong.  It was easy to conclude which of the above answers is more likely.

Actually, I think, but I'm not certain, that a roulette wheel has a null spot, neither red nor black.  If the ball lands in that spot, nobody wins.  So the odds of choosing correctly red or black are somewhat lower than 50-50.  This is how the casino makes their money, over the long run.

.

Dec 14, 2020
#2
+2155
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Merely Unlucky?  No! Not merely. This unlucky series of failures is astronomically unlikely.

Ignoring the roulette wheel for the moment, this question has the same probability as flipping a fair coin 59 times and incorrectly guessing the outcome every time –or correctly guessing the outcome every time. It’s also the same probability as having 59 heads appear or 59 tails, or...

For this question, a failure is a statistical success.

With a 50% probability of success/fail, the probability of 59 sequential failures is (0.5)^59 is:

$$\dfrac{1}{576460752303423488} = 1.73472347597680709441192448139190673828125E-18$$

Here are some perspectives comparing other probabilities to the probability of 59 sequential failures in guessing coin flips.

Lotteries: PowerBall, MegaMillions, and Keno 80/20

The probability of simultaneously winning both jackpots in the PowerBall and MegaMillions lotteries with a single ticket for each lottery is 6.5 times greater than (59) sequential failures in guessing coin flips.

The Math:

$$\dfrac {1}{292201338} * \dfrac {1}{302575350}= \dfrac {1}{88412922115818300}$$

$$\dfrac{576460752303423488}{88412922115818300} \approx 6.52$$ times more likely than (59) sequential failures in guessing coin flips.

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Keno 80/20

Matching all 20 numbers on Keno 80/20 is 163 times more likely than (59) sequential failures in guessing coin flips.

The Math:

Probability of matching all 20 numbers on Keno:

$$\dfrac {1} {\dbinom{80}{20}} = 3535316142212174320$$

$$\underbrace {576460752303423488}_{59 \;sequential \;coin \;fails} \div \underbrace {3535316142212174320 } _{20 \; Keno \; matches} \approx 163.13 \text{ times more likely}$$

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Blackjack:

The probability of 13.4064 sequential (natural) BlackJack wins is slightly less than the probability for (59) sequential failures in guessing coin flips.

Blackjack Math:

The probability of a single deck (natural) Blackjack (Blackjack on the initial deal of two (2) cards) is

$$2* \dfrac {4}{13}* \dfrac {4}{51} = \dfrac {8}{169} \approx 0.047619\\$$

Inverse of  $$\dfrac {8}{169} = 21.125$$  (This is used as the base of a Logarithm)

$$\log_{21.125} (576460752303423488) \approx 13.4064$$ sequential Blackjack wins for every (59) sequential failures in guessing coin flips.

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Royal Flush:

The probability of 3.0555  sequential (natural) Royal Flush wins in 5-card draw poker will occur slightly more than 3 times before (59) sequential failures in guessing coin flips.

Royal Flush Math:

The probability of a (natural) Royal Flush for 5-card draw poker is $$\dfrac {1}{649740}$$

This gives

$$\large log_{649740}(576460752303423488) =3.0555$$ sequential Blackjack wins for every (59) sequential failures in guessing coin flips.

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Testing the sample space:

Conducting this experiment 576,460,752,303,423,488 times gives an expectation of one (1) statistical success.   However, expectation is not probability. The probability of realizing this expectation is:

$$1- \left(\frac {576460752303423487}{576460752303423488}\right )^{576460752303423488} \approx 63.21 \%$$

So the probability of NOT realizing this expectation is:

$$\approx 36.79 \%$$ (Not coincidentally, this is equal the inverse of Euler's Number (e) 2.71828...)

Yep, doing this experiment 576,460,752,303,423,488 times will result in failure well more than $$\dfrac {1}{3}$$ of the time.

How long to conduct an experiment?

Figure it will take about 40 seconds, with an experienced croupier spinning the (special, no green pocket zero(s)), red-black, even-odd, Roulette wheel and ball; and the gambler flipping the coin and placing the wager while the ball orbits the wheel.  That’s just one spin in the experimental attempt. The flipping and spinning continues until the gambler matches (wins) (or fails to match (loses) 59 times in sequence). If the gambler wins, the experiment begins again.   Most of the experiments will fail before the fifth flip and spin.  Remember, 576,460,752,303,423,488 is the number of experiments, NOT the number of coin flips and wheel spins, which will be many multiples.

So how long would it take to do 576,460,752,303,423,488 coin flips and wheel spins, figuring 40 seconds for each one?

That’s 23,058,430,092,136,939,520 seconds. That’s a long time. It’s (53) times the age of the universe. Calculated using the Big Bang event as a starting point –the formation/creation of the universe –more formally, the starting point when the universe inflated and space-time began. (This should not be confused with the big bang of your mum and pop, the starting point that procreated you, even though that was also long ago.)

So Ron, do you still think the gambler is Merely Unlucky?

If you do, I’m willing to bet a Rouble that some Russian fucked with the Roulette wheel?

Known in Russia as Гребаная русская рулетка

GA

Dec 19, 2020
edited by GingerAle  Dec 19, 2020
#4
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I put this in to wolfram alpha 1 - (576460752303423487/576460752303423488)^576460752303423488 an d it says it = 0 but you put it = 63.21

https://www.wolframalpha.com/input/?i=1-+%28%28576460752303423487%29%2F%28576460752303423488%29%29%5E576460752303423488

Guest Dec 21, 2020
#6
+2155
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Well, I am disappointed that the world’s most advanced computational engine (accessible to the general public) fails to solve this directly.

But...as I’ve often said, “to properly use a computer, one has to be smarter than the computer; and we Genetically Enhance Chimps are... So, here’s how to sweet-talk the hybrid WolfRam computer into giving you the correct solution.

First, input this (576460752303423487/576460752303423488), then copy and use the decimal form as the base of the exponent:

1- (0.99999999999999999826527652402319290558807551860809326171875)^ 576460752303423488

The Wolf will return: 0.6321205588285576787235607813032040042372304533972097152888327911...

(It will offer up to 4157 decimals)

https://www.wolframalpha.com/input/?i=1-+%280.99999999999999999826527652402319290558807551860809326171875%29%5E+576460752303423488

GA

GingerAle  Dec 22, 2020
#3
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Hi Ginger.  It's always a delight to happen across one of your postings.  I'm sure it didn't escape you that I said "merely very unlucky" was the best answer, not in an absolute sense, but in comparison to the other five choices.  All you did was criticize my reasoning ... you neglected to say which choice you think would be better, perhaps that wasn't an oversight.

The "e." answer does include the word "very" which, as you've demonstrated – I'm assuming your calculations are accurate, or at least, accurate enough – is an apropos modifier.  Maybe it should have been in an emphasized typeface, but I'm not the one who posted the question.

I don't play poker often.  In fact, practically never.  But one day I was playing with my daughter and son-in-law, and two teenage grandchildren, using macaroni noodles as chips – uncooked noodles, of course – and I did get a Royal Flush.  No joker, no wild cards at all, just the 10 thru Ace of hearts, and I didn't draw to it, I received it on the deal.  So, unlikely is not synonymous with impossible.

Goodnight, Gracie.

.

Dec 19, 2020
#5
+2155
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Hi Ron,

I'm sure it didn't escape you that I said "merely very unlucky" was the best answer, not in an absolute sense, but in comparison to the other five choices.

Actually, you didn’t say that, but I understood that was part of your reasoning.

All you did was criticize my reasoning

Wow! You are touchy!

Your reasoning was worthy of criticism, but what I did was give an opinion, supported with math-based comparisons to probabilities of familiar games of chance. The math comparisons articulate the ratios or the equivalent number of sequential statistical successes for unlikely and extremely unlikely events.  While these articulations contradict your reasons for selecting “merely very unlucky,” they do not overtly criticize your reasoning.   I’m sure you’ve read enough of my posts to know when I’m criticizing.  (I’m still LMAO over the Gracie Allen comment. –It’s my best one-liner troll post ever.)

...Continuing with the postscript assessment:

My introduction to the post,

Merely Unlucky?  No! Not merely. This unlucky series of failures is astronomically unlikely,

is my articulation of disagreement for the use of the word “merely,which was intentionally used by the teacher who wrote this question to trivialize or down play the “very unlucky” part of the phrase. The word “simply” could be substituted for “merely” as it is used in this context.

Simple things happen frequently, complex things happen less frequently.  But there is nothing simple about a series of events, that when aggregated, is so statistically unlikely that before it happens the Earth’s sun will be a cold brown dwarf (if it still exists at all), the Milky Way galaxy will with merge with the Andromeda galaxy, a team from the PeeWee football league will win the Super Bowl, and CPhill will have extracted the Roman Zero from Sisyphus’s Bolder.

The "e." answer does include the word "very" which, as you've demonstrated – I'm assuming your calculations are accurate, or at least, accurate enough – is an apropos modifier.  Maybe it should have been in an emphasized typeface, but I'm not the one who posted the question.

There are probably no words typically used as modifiers in English, Latin, Greek, or any language that can conveniently depict the absurd statistical improbability of this hypothetical question, – at least, not without sounding hysterical.

Here’s an analogy:

To bottle all the water in the Atlantic Ocean requires a very large number of gallon jugs.   As you may see, though true, it’s an understatement.  It’s absurd.

To bottle all the water in the Atlantic Ocean merely requires a very large number of gallon jugs. Adding ‘merely’ makes it absurdly funny.

The modifiers in this answer are intentionally used to teach novice students about highly improbable events.

...I'm assuming your calculations are accurate, or at least, accurate enough...

They are, but for most novice students of statistics, the path will be a long one before they begin to understand the significance.   While many can at least intuit an understanding for large ratios, the enormity of the numbers in the ratios is lost to them. It’s worse with the sequential successes of highly improbable events. For example they see the royal flush and think “Oh, I only need three (3) of these in a row. Three is a low number –that shouldn’t be too difficult to do.”

Most students will not realize that a professional poker player will likely never have a natural royal flush in a five-card poker game, and will only see two, in their entire careers.  And in a billion billion life times they will never see two in a row, let alone three.

...and I did get a Royal Flush.  No joker, no wild cards at all, just the 10 thru Ace of hearts, and I didn't draw to it, I received it on the deal.

Congratulations. It’s too bad you only won macaroni!

Were you playing five-card or seven-card poker?   It’s (1/649740) for five-card and (1/30940) for seven-card poker. The seven-card is (21) times more likely, but still amazing. In my logbook where I’ve recorded over 29,000 poker games, I’ve never had a natural royal flush. I’ve had a natural king-high straight flush twice, and one queen high straight flush. It’s worth noting that in seven-card poker the royal flush is slightly more likely to occur than the other straight flush hands. In five-card poker, the probability is the same for any given high-card.

So, unlikely is not synonymous with impossible.

You’re right, it’s not.

What infinitesimal number is small enough to be synonymous with zero?

How small does a probability have to be before it becomes synonymous with impossible?

What’s the probability that a single shuffle of a randomized standard deck of 52 cards returns to its factory-sealed box order of A-K spades, A-K hearts, A-K clubs, and A-K diamonds?

Shuffling the cards once per second, how long would it take before the expectation of (1) success reaches a probability of (63.21%)?  At this point in time, how many statistical successes will our coin-flipping-roulette player have?

$$\large \log_{576460752303423488}(52!) \approx 3.82$$ statistical successes

So if our coin-flipping-roulette player takes forever to have one success, then our shuffler will take (3.82) forevers.

-------

you neglected to say which choice you think would be better, perhaps that wasn't an oversight.

You are right, it wasn’t an oversight: My offer “...to bet a Rouble that that some Russian fucked with the Roulette wheel,” implies answer (b).

I’m still willing to make the wager. A Russian Rouble is 0.014USD; that’s about 8 elbow macaroni noodles.  You know, my point in making the wager was to encourage you to use your noodle –to think outside of the box (of macaroni).

GA

Dec 22, 2020
edited by GingerAle  Dec 22, 2020