Crows might understand probabilities

Researchers at the University of Tübingen are studying crows’ abilities to understand statistical inference. For Ars Technica, Kenna Hughes-Castleberry reports:

To do this, Johnston and her team began by training two crows to peck at various images on touchscreens to earn food treats. From this simple routine of peck-then-treat, the researchers significantly raised the stakes. “We introduce the concept of probabilities, such as that not every peck to an image will result in a reward,” Johnston elaborated. “This is where the crows learn the unique pairings between the image on the screen and the likelihood of obtaining a reward.” The crows quickly learned to associate each of the images with a different reward probability.

In the experiment, the two crows had to choose between two of these images, each corresponding to a different reward probability. “Crows were tasked with learning rather abstract quantities (i.e., not whole numbers), associating them with abstract symbols, and then applying that combination of information in a reward maximizing way,” Johnston said. Over 10 days of training and 5,000 trials, the researchers found that the two crows continued to pick the higher probability of reward, showing their ability to use statistical inference.

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Changes to Blackjack payouts so that gamblers lose more to casinos

Katherine Sayre, for The Wall Street Journal, on Las Vegas casinos squeezing out more juice:

Gambling companies such as MGM Resorts and Caesars, the two biggest operators on the Strip, have reduced how much they will pay for winning hands at blackjack at many of their tables, data show.

Blackjack, a fast-paced card game, historically paid out a ratio of 3:2 when a player hit 21 on the first two cards. That means a gambler wins $15 for every $10 bet. Now, many blackjack tables on the Strip pay out at 6:5, which means that same $10 yields only $12.

During my very first advanced probability course in college, the professor went right into the math of “the house always wins.” It only takes a tiny house advantage to guarantee you lose all your money over a long enough period of time.

For example, it would seem that if you play Blackjack following the same strategy as the dealer, the odds of you winning and losing would be even. But the dealer will always take your money if you bust, regardless of what hand they get. That’s enough advantage for the casino. Well, it was enough, until they started paying out less and increasing minimum bets.

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Impossible or improbable lottery results

There was a government-run lottery in the Philippines with a $4 million jackpot, and two improbable things happened. First, the numbers selected were all multiples of nine: 9, 45, 36, 27, 18, and 54. Second, 433 people won. The natural reaction from the public was that something sketchy happened, especially since the government has a history of sketchiness.

However, as statisticians and mathematicians do when rare and improbable events occur, they setup hypotheses and calculate probabilities. Terence Tao calculated the odds and noted that the lottery outcome was a highly improbable event under certain assumptions. But:

So this clearly demands some sort of explanation. But in actuality, many purchasers of lottery tickets do not select their numbers completely randomly; they often have some “lucky” numbers (e.g., based on birthdays or other personally significant dates) that they prefer to use, or choose numbers according to a simple pattern rather than go to the trouble of trying to make them truly random.

Nine happens to be a lucky number in some cultures. Also, as Tao notes, the multiples of nine form a diagonal line on the physical lottery ticket, which could lend to more people just going with simple geometry.

The chances of each winning number being a multiple of nine is improbable, but any other individual number selection is equally improbable.

So if you assume one improbable event, the winning lottery numbers, paired with a less improbable event, the players’ selection of their own numbers, it doesn’t seem that unbelievable, statistically speaking.

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Introduction to Probability for Data Science, a free book

Introduction to Probability for Data Science is a free-to-download book by Purdue statistics professor Stanley H. Chan:

We need a book that balances the theory and practice. We need a book that provides insights and not just theorems and proofs. We need a book that motivates the students, telling them why probability is so essential to their work. We need a book that highlights the impacts of the subject. From over than half a decade of teaching the course, I have distilled what I believe to be the core of probabilistic methods. I put the book in the context of data science, to emphasize the inseparability between data (computing) and probability (theory) in our time.

Download a free PDF copy or buy a physical copy.

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Odds of winning the big Mega Millions prize

With tonight’s Mega Millions jackpot estimated at $1.28 billion, you might be wondering what the odds of winning are, even if you know the chances are super slim for an individual. (On the other hand, the more tickets purchased overall, the greater the chances that someone in the country wins.) For The Washington Post, Bonnie Berkowitz and Shelly Tan made a playful quiz to test your perception of 1 in 302.6 million.

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Calculating win probabilities

Zack Capozzi, for USA Lacrosse Magazine, explains how he calculates win probabilities pre-game and during games. On interpretation, which could easily apply to other sports and all forecasts:

But interpretation here matters quite a bit. And this is frustrating for some people, but that 61 percent should be interpreted as: “if these teams played 100 times, we would expect Marquette to win 61 of those games.” It definitely does not mean that the model is 61 percent confident that Marquette will win.

This is a bit odd, but this also means that if the Win Probability model gives Team A a 90% chance to beat Team B, there is nothing wrong with the model if Team B ends up winning the game. The issue would arise if, out of 100 90-percent win probability games, the favorite wasn’t winning around 90 of those games. When the model says 90 percent, you want it to mean 90 percent.

I wonder how many people incorrectly interpret the probability as “61 percent confident”. I bet a lot.

I do know that ever since the Golden State Warriors lost to the Cleveland Cavaliers in the 2016 NBA Finals — while holding a 90-something percent win projection by FiveThirtyEight — I stopped paying attention to win probability. But learning more about the calculation made it more interesting.

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Looking for similar NBA games, based on win probability time series

Inpredictable, a sports analytics site by Michael Beuoy, tracks win probabilities of NBA games going back to the 1996-97 season. When a team is up by a lot, their probability of winning is high, and then flip that for the losing team. So for each game, you have a minute-by-minute time series of win probability.

Beuoy added a new feature that looks for games with similar patterns a.k.a. “Dopplegamers”.

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Probability you will break up with your partner

Rosenfeld, et al. from Stanford University ran a survey in 2009 for a study on How Couples Meet and Stay Together. Dan Kopf and Youyou Zhou for Quartz used this dataset to estimate the probability that you will break up with your partner, given a few bits of information about your current relationship.

The Stanford data page says a 2017 release is on the way. I’m curious how, if anything, has changed in relationships between 2009 and now.

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