Category: probability

What Are The Odds Of Getting Bit By Both A Bea…

What Are The Odds Of Getting Bit By Both A Bear And A Shark?

“For someone who doesn’t surf in tropical waters and doesn’t go backcountry hiking and camping where bears are prevalent, it’s true: your odds are trillions-to-one that you’ll get both bitten by a bear and a shark in your lifetime. But behavior and risk-exposure matter. It’s not surprising news when someone gets bit by a snake: it happens about 8,000 times per year in the USA alone. It’s not surprising when a surfer gets bit by a shark; surfers are the most likely people to receive shark bites and it happens dozens of times a year. And it’s not surprising to encounter a hungry black bear in the back country woods. And finally, it’s not surprising to survive all of these, as it’s very uncommon for any of these encounters to be fatal.

Dylan is certainly an unusual case, but in every case, he put himself in the most at-risk group for these types of encounters.”

Earlier this week, it was reported that a young man named Dylan McWilliams was bitten by a shark while surfing in Hawaii. This wouldn’t be such a big deal on its own, but last year Dylan was bitten by a bear while camping in Colorado, and two years before was bitten by a snake. Is he just the unluckiest person on Earth, who overcame astronomically small odds to have all three of these things happen to him? Or are the odds, given his behavior and location and circumstances, far higher than a naive calculation would indicate?

What happened to Dylan was unusual, but it’s not nearly as unlikely as you might think. Come get a solid lesson in conditional probability today! (It’s more fun than it sounds!)

He’s On Fire! How The Hot Hand Helped Golden State Become…

He’s On Fire! How The Hot Hand Helped Golden State Become NBA Champions

“When you do your statistics (and your averaging) correctly, you find that many NBA players really are streakier shooters than others. Klay Thompson may be the streakiest shooter of all, but Steph Curry, LeBron James, and Kyrie Irving all “get hot” in a streaky way that goes beyond what their normal shooting percentages would indicate. A player getting hot at the right time can help their team tremendously, and now that we’re finally doing our statistics right, we’re finding something surprising but robust: for at least some NBA players, the effect of the hot hand is real.”

When someone makes a shot, it’s a little instinctive to want to give them the ball again to see if they’ll make another. If they make three or four in a row, you’ll really want to ride that ‘hot hand’ as far as it will go. And if your teammate has made six, seven, or eight shots in a row, you know you’re beginning to witness something very special. There’s nothing like getting hot in quite that way. Except, for decades, statisticians have been telling us that the hot hand is nothing more than a fallacy, the way we convince ourselves that you can better predict the results of a coin flip or a roulette wheel based on prior results. This was based in solid enough numbers, but there was a flaw: by selecting for streaks in the first place, the scientists doing the analysis had restricted their choices! When you account for that restriction, it turns out that the “hot hand” is very real.

NBA Jam had it right all those years ago. For some players, they really do go into an “on fire” mode that makes them streakier – for better and for worse – than others.

Isaac Newton vs. Las Vegas: How Physicists Used Science To Beat…

Isaac Newton vs. Las Vegas: How Physicists Used Science To Beat The Odds At Roulette

“By 1961, Thorp and Shannon had built and tested the world’s first wearable computer: it was merely the size of a cigarette pack and able to fit into the bottom of a specially-designed shoe. Toe switches would activate the computer once the wheel and ball were set into motion, collecting timing data for both. Once the computer calculated the most likely result, it would transmit that value as musical tones to a tiny speaker lodged in an earpiece. The wires were camouflaged as much as possible.”

Did you know the world’s first wearable computer was built all the way back in the 1960s, was worn on your feet… and was used to help gamblers cheat at roulette? Physicists and mathematicians work with probability and predicting the behavior of a given system a lot, and when you combine that with the science of simple motion (as on a roulette wheel), the possibility of ‘beating the odds’ suddenly becomes real. Security measures that seem commonplace today in casinos, such as roulette wheels with no observable defects, a ban on computers and ‘table talk,’ and the inability to place late bets, all came about because of how scientist/gamblers have successfully beaten the house in the past.

From the 1940s up to the modern day, come hear the story of how simple physics helped defeat the casinos, and how the saga, for a few people, is still ongoing today!

fuckyeahphysica: This shows that the probability of a random variable is maximum at the average and…

fuckyeahphysica:

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This shows that the probability of a random variable is maximum at the average and diminishes as one goes away from it, eventually leading to a bell-curve.

Bean Machine / Galton Box

Why does rolling balls down a plane form a gaussian distribution ? Well you see, as the balls are coming down they hit these pins.

When they hit one, they either go left or they go right.

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Now if you sort of track the ball as it makes it way down to the bin below, then you will notice that :

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Here’s another iteration:

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Generalization

Given that I have ‘k’ rows of pins, then what is the probability that I will find a ball in say bin ‘2′ ? Well, if you trace the path from the previous diagram then you will see that there are 3 ways to get there.

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If you work it out for the other bins as well, you will end up with:

k = 0                 (1)

k = 1                (1,1)

k = 2               (1,2,1)

k = 3              (1,3,3,1)

.

.

.

I could go on but I guess you have already found the pattern out – Pascals’ triangle/ Binomial coefficients.

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From the pascal’s triangle, you can find the probability in any bin!

That’s the foundation on how the bean machine and binomial/Gaussian distribution are related.  To quench your thirst for more knowledge look at the reference section of this wikiThe Bean Machine

Have a great day!

** But what if the probabilities were NOT the same ? – Ask this to yourself and this will help you generalize things!

This shows that the probability of a random variable is maximum at the average and diminishes as one…

image

This shows that the probability of a random variable is maximum at the average and diminishes as one goes away from it, eventually leading to a bell-curve.

The odds of your unlikely existence were not infinitely…

The odds of your unlikely existence were not infinitely small

“But there’s a fun, important, and underappreciated consequence of Bayes’ theorem that can tell us something vital about any of these steps: the odds of any one of them happening, no matter how small, could not have been infinitesimal. If you want to create our Universe with our laws of physics, our local group, our Sun, our Earth, and every one of us, given all the conditions that existed before the Big Bang, that probability may be very, very small, but it can’t be infinitely small. If it were, our model for the conditions that existed before the Big Bang could be ruled out immediately, with no need to gather data or make measurements.”

There are a great many events that occurred to give rise to the world, the Universe, and you. Everything from the Big Bang to the existence of the laws of physics to the cosmic history that created Earth to the biological history that gave rise to the 7+ billion of us today needed to unfold exactly as it did in order for things to be the way they are today. It’s an exceedingly unlikely story, and yet the fact that we’re here is evidence that it happened exactly in this fashion. But no matter how unlikely it appears, we can be 100% certain that our arrival at this point in time with these exact conditions wasn’t infinitely unlikely. In fact, it’s an inescapable consequence of Bayes’ theorem that the existence of things as they are today implies it had a finite, not infinitesimal, probability of turning out this way.

No matter how unlikely every event may have been in the past, the odds weren’t infinitely small. Find out how we know today!