Showing posts with label gambling. Show all posts
Showing posts with label gambling. Show all posts

Monday, December 7, 2020

How to (maybe) win at dice

Does God play dice with the universe?

Imagine I gave you an ordinary die, not special in any way, and I asked you to throw the die and record your results (how many 1s, how many 2s, etc.). What would you expect the results to be? Do you think you could win by choosing some numbers rather than others? Are you sure?

(Image source: Wikimedia Commons. Author: Diacritica. License: Creative Commons.)

What you might expect

Let's say you thew the die 12,000 times, you might expect a probability distribution something like this. This is a uniform distribution where all results are equally likely.

You know you'll never get an absolutely perfect distribution, so in reality, your results might look something like this for 12,000 throws. 

The deviations from the expected values are random noise which we can quantify. Further, we know that by adding more dice throws, random noise gets less and less and we approach the ideal uniform distribution more closely. 

I've simulated dice throws in the plots below, the top chart is 12,000 throws and the chart on the bottom is 120,000 throws. The blue bars represent the actual results, the black circle represents the expected value, and the black line is the 95% confidence interval. Note how the results for 120,000 throws are closer to the ideal than the results from 12,000 throws.


What happened in reality - not what you expect

My results are simulations, but what happens when you throw dice thousands of times in the real world?

There's a short history of probability theorists and statisticians throwing dice and recording the results.
  • Weldon threw 12 dice 26,306 times by hand and sent the results to his friend Francis Galton.
  • Iversen ran an experiment where 219 dice were rolled 20,000 times.
Weldon's data set is widely used to illustrate statistical concepts, especially after Pearson used it to explain his \(\chi^2\) technique in 1900. 

Despite the excitement you see at the craps tables in Las Vegas, throwing dice thousands of times is dull and is, therefore, an ideal job for a computer. In 2009, Zachariah Labby created apparatus for throwing dice and recording the scores using a camera and image processing. You can read more about his apparatus and experimental setup here. He 'threw' 12 dice 26,306 times and his machine recorded the results. 

In the chart below, the blue bars are his results, the black circle is the expected result, and the black line is the 95% confidence interval. I've taken the results from all 12 dice, so my throw count is \(12 \times 26,306\).
This doesn't look like a uniform distribution. To state the obvious, 1 and 6 occurred more frequently than theory would suggest - the deviation from the uniform distribution is statistically significant. The dice he used were not special dice, they were off-the-shelf standard unbiased dice. What's going on?

Unbiased dice are biased

Take a very close look at a normal die, the type pictured at the start of this post which are the kind of die you buy in shops.

By convention, opposite faces on dice sum to 7, so 1 is opposite 6, 3 is opposite to 4, and so on. Now look very closely again at the picture at the start of the post. Look at the dots on the face of the dice. Notice how they're indented. Each hole is the same size, but obviously, the number of holes on each face is different. Let's think of this in terms of weight. Imagine we could weigh each face of the dice. Let's pair up the faces, each side is paired with the face opposite it. Now let's weigh the faces and compare them.

The greatest imbalance in weights is the 1-6 combination. This imbalance is what's causing the bias.

Obviously, the bias is small, but if you roll the die enough times, even a small bias becomes obvious. 

Vegas here I come - or not...

So we know for dice bought in shops that 1 and 6 are ever so slightly more likely to occur than theory suggests. Now you know this, why aren't you booking your flight to Las Vegas? You could spend a week at the craps tables and make a little money.

Not so fast.

Let's look at the dice they use in Vegas.

(Image source: Wikimedia Commons. Author: Alper Atmaca License: Creative Commons.)

Notice that the dots are not indented. They're filled with colored material that's the same density as the rest of the dice. In other words, there's no imbalance, Vegas dice will give a uniform distribution, and 1 and 6 will occur as often as 2, 3, 4, or 5. You're going to have to keep punching the clock.

Some theory

Things are going to get mathematical from here on in. There won't be any new stories about dice or Vegas.

How did I get the expected count and error bars for each dice score? Let's say I threw the dice \(x\) times, it seems obvious we would get an expected count of \(\frac{x}{6}\) for each score, but why? What about the standard error?

Let's re-think the dice as a Bernoulli trial. Let's choose a score, say 1. If we throw the dice and it shows a 1, we consider that a success. If it shows anything else, we consider it a failure. Because we have a Bernoulli trial, we can use the binomial distribution to model the results.

  • \(n\) is the number of throws
  • \(p\) is the probability of getting a 1, which is \(\frac{1}{6}\)
  • \(q = 1- p\) is the probability of getting 2-6, which is \(\frac{5}{6}\)
So, again using Wikipedia's handy summary, for \(n\) throws:
  • The mean is \(np = 12 \times 26,306 \times \frac{1}{6} = 52,612\)
  • The standard deviation is \(\sqrt{npq} = \sqrt{12 \times 26,306 \times \frac{1}{6} \times \frac{5}{6}} = 209.388\) 
  • The 95% confidence interval is \(52,202 \) to \(53,022\) (standard deviation by 1.96).

Publications

Academics live or die by publications and by citations of their publications. Labby's work has rightly been widely cited on the internet. I keep hoping that some academic will be inspired by Labby and use modern robotic technology and image recognition to do huge (million-plus) classical experiments, like tossing coins or selecting balls from an urn. It seems like an easy win to be widely cited!

Monday, September 21, 2020

The gamblers' fallacy

An offer you can't refuse?

Imagine you're in a casino playing craps, a game where you bet on the outcome of two dice thrown at the same time. The probability of a double six coming up is 1/36, but no-one has thrown a double six for over 110 throws. The table is starting to get crowded and noisy with people betting on a double six. It's due to come up, and it must come up soon. 

(Still no double six. Source: Wikimedia Commons. License: Creative Commons. Author: Gaz.)

A new player rolls the dice; snake-eyes (double ones) - still no double six.

You feel a tap on your elbow. A lady in a cocktail dress whispers to you that she'll give you odds of 20 to 1 for a double six.

Another player rolls the dice; easy-four (one and three) - the expectation for a double six mounts.

Your new friend whispers that she'll reduce the odds soon; she asks if you want to take the bet.

It's now 130 throws since a double six has occurred and it should have occurred 3 or 4 times by now.

Do you take the bet?

The gambler's fallacy 

The gambler's fallacy is the belief that the outcome of a random event is somehow influenced by the previous random events. In our craps case, some examples might be:

  • double six hasn't come up in 130 throws, so it's much more likely to come up now (the probability is higher than 1/36)
  • double one has just come up, therefore it's not likely to come up again soon (the probability is less than 1/36).

It's a fallacy because each roll of the dice is completely independent; it doesn't matter what the previous throws were. There could have been 1,000 throws without a double six, but the probability of a double six will always be 1/36. The same thing for the snake-eyes example, if a snake-eyes has been thrown, the probability of throwing another snake-eyes immediately after is still 1/36.

Let me lay this out even more starkly, in craps:

  • At the very first roll of the dice, the probability of a double six is 1/36.
  • After ten rolls of the dice, the probability of the next roll being a double six are 1/36.
  • After 100 rolls without a double six, the probability of the next roll being a double six is 1/36.
  • After 200 rolls without a double six, the probability of the next roll being a double six is 1/36.
  • After 1,000 rolls without a double six, the probability of the next roll being a double six is 1/36.

Otherwise rational people are fooled by the gambler's fallacy all the time. As the money increases and the emotion heightens, the gambler's fallacy becomes easier and easier to fall for, as we'll see.

The Italian lottery

The story starts in Venice, Italy in May 2003. The Venice lottery was a game where 6 numbered balls (plus a bonus ball) were selected from a set of 90 numbered balls. The lottery was run twice a week. Each number should come out on average once every 7-8 weeks. As with all government-sponsored lotteries, the results were well-publicized.

In May 2003, the number 53 came up. Then it didn't come up again.

By October, people realized the number 53 was overdue. They started to gamble on 53 occurring - it was overdue, so it must come up. But 53 just didn't come up.

News of the 53 drought started to spread, and more and more Italians started to bet that 53 would occur, but it didn't. It didn't come up in November or December either. 

In January of 2004, a woman from Carrara committed suicide because she'd spent her family's life-saving gambling that 53 would come up. It didn't.

Still, 53 didn't come up.

People went crazy betting money that 53 would come up, they became known as '53 addicts'. They were sure it must come up. Sadly, it didn't. A man from Signa shot his wife, his son, and himself after losing money gambling on 53.

Still, 53 didn't come up.

Italians gambled and lost a huge amount of money on 53, an estimated 4 billion Euros. They had fallen for the gambler's fallacy and believed that 53 must come up soon.

Eventually, 53 did come up - in February 2005, after 182 draws (remember, each draw was seven balls).

The Venice lottery made a lot of money, but the Italian gamblers did not.

How the cocktail dress lady (and casinos) makes money

To understand if the cocktail dress lady was offering a good deal, we need to relate probability to odds.

The probability of a double six is 1/36.

The odds are the ratio of the probability the event will occur divided by the probability the event will not occur:

\[odds = \frac{P}{1-P}\]

The odds of a double six are:

\[ odds_{66} = \frac{\frac{1}{36}}{\frac{35}{36}} = \frac {1}{35}\]

which a bookie might quote as 35 to 1.

Generally speaking, casinos and bookies make money in one of two ways:

  • The probabilities don't add up to 1.
  • They rely on the gamblers' fallacy and offer worse odds than a fair analysis would suggest.

Let's imaging there are ten horses in a race. Each horse has a 10% chance of winning, which are odds of 9 to 1. If you win, you get your stake money back, so a winning bet of $1 gives you $10. If ten punters bet $1 on each horse, the bookie takes $10, but one of the horses must win, so the bookie pays out $10. 

(Bookmakers make money. You don't. Image source: Wikimedia Commons. License: Creative Commons. Author: Grand Island Tourism )

To make money, the bookie reduces the odds. Instead of offering 9 to 1 on each of the horses, the bookie offers 8 to 1. The bookie still takes in $10, but this time only pays out $9. In the real world, it's more complicated, but you get the idea.

The other way to make money is to underprice probabilities. A double six should be offered at 35 to 1, but you could offer it at 20 to 1. This is a horrible deal, but if gamblers have a bad case of the gambler's fallacy, they may be convinced the probability is much higher than 1/36 and they may even view a horrible deal as the deal of a lifetime. The casino, or the lady in the cocktail dress, make money by knowing the odds and knowing when to offer a deal that seems attractive, but isn't.

Not only should you not accept the 20 to 1 offer, you should offer it to other players.

Gambler's fallacy in Reno, Nevada and Monte Carlo

Obviously, there are naive gamblers in Las Vegas, but do people really fall for the gamblers' fallacy at the roulette table? After all, you have to have some level of sophistication to understand and play the game, so surely gamblers are savvy and know how to price bets appropriately? It seems that they don't always.

Using videotape data supplied by a casino in Reno, Nevada, two researchers tracked the pattern of gambling on roulette. If gamblers have fallen for the gambler's fallacy, you might expect to see certain patterns of betting, for example, if red hasn't come up as often as expected, they might bet more on red. The researchers found small, but significant examples of the gambler's fallacy The reality is then, there are people who fall for the fallacy, even those playing a sophisticated game like roulette.

(Image source: Wikimedia Commons. License: Creative Commons. Author: Ken Lund.)

Another object lesson in the gambler's fallacy occurred at a roulette table in a casino, this time at a casino in Monte Carlo. In 1943, the ball landed on red 32 times in a row. The people who thought black must come up were cleaned out. 

The gamblers' fallacy elsewhere

The gambler's fallacy has been an active area of research for some time. Variations of it have been found in different places:

Let's imagine you're an asylum judge. You're aware of the average 'success' rate for applicants and you don't want to be too far from the average. Let's assume the cases are randomly assigned (deserving and undeserving). By random chance, you might get a long string of deserving or undeserving cases, maybe as many as twenty in a row. The gambler's fallacy may kick in after a series of similar cases, for example, the first ten cases were deserving, so the eleventh 'must' be undeserving, as a result, you judge more harshly based on expectation.

The gambler's fallacy in business

If you listen closely enough, you hear business people make the gambler's fallacy all the time. How often have you heard these kinds of phrases:

  • We've won the last 8 contracts, so we must win the next one.
  • We just failed to land the last 6 deals, so the odds of us landing the next deal are high.

Despite what people say, business can be strongly driven by belief and not rationality. If everyone needs a deal to be landed, then the collective view might become that a deal will be landed, regardless of what a realistic measure of the probabilities are.

How to guard against the gamblers' fallacy

There's something about humanity and our (mis)understanding of statistics that makes us vulnerable to the gambler's fallacy. The best teacher might be experience. How many Italians who bet on 53 would do so again? There's some evidence that the gambler's fallacy is particularly strong when the data evolves over time, which ties in with the Italian lottery and casino examples. Perhaps the best defense is to take a step back and view the data as a whole, then make a decision away from the influence of others.

The existence of opulent casinos should be a lesson that those who understand probability can make money from those who do not.