The subject of coin tossing keeps coming up, no doubt because it is something the average person can relate to--no abstruse definitions like with limits (in Calculus). And that forces me to go back and find references for some of the basic points in the arguments that should be known but aren't. I've decided it's about time to accumulate the basic points and along with references:
The probability of flipping heads on a coin is not 1/2. The assumption that flipping heads on a coin is 1/2 is a mathematical model and not reality which is akin to using 3.14 for pi. Coin tossing is a deterministic process in physics as demonstrated by a coin tossing machine, "To make his point, Diaconis commissioned a team of Harvard technicians to build a mechanical coin tosser -- a 3-pound, 15-inch-wide contraption that, when bolted to a table, launches a coin into the air such that it lands the same way every single time. Diaconis himself has trained his thumb to flip a coin and make it come up heads 10 out of 10 times. But what he really wanted to know was whether unrehearsed tosses -- by ordinary folk who flip coins with unpredictable speeds and heights and catch them at different angles -- would show that the outcome of the act was, in fact, random." Persi Diaconis, Susan Holmes, and Richard Montgomery are authors of the article "Dyanmical Bias in the Coin Toss" (.pdf). There is a Numberphile video with Diaconis (about 8 minutes) that gives a brief overview and there is a YouTube lecture by Diaconis (about 55 minutes) with more detail. One of the main assumptions is that you start the coiin with the heads side up is
Mathematician William Feller was a well known expert in probability who wrote a classic book An Introduction to Probability Theory and Its Applications in which you can find (by click on "Look Inside") the following quote on page 19: "As a matter of fact, whenever refined statistical measures have been used to check on actual coin tossing, the result has invariably been that head and tail are not equally likely. And yet we stick to our model of an "ideal" coin even no good coins exist. We preserve the model not merely for its logical simplicity, but essentially for its usefulness and its applicability.".
The coin flipping model has two assumptions built into it:
- There are two outcomes (heads and tails)
- The two outcomes are equally likely.
The first assumption isn't always true. The Abstract of paper (by Murray and Teare) mentions the odds of an American nickel landing on its edge is about 1/6000.
The deterministic nature of coin flipping can be found in the Phys.org article "Heads or tails? It all depends on some key variables" which says:
"But first, here's what the researchers concluded: Using a high-speed camera that photographed people flipping coins, the three researchers determined that a coin is more likely to land facing the same side on which it started. If tails is facing up when the coin is perched on your thumb, it is more likely to land tails up. How much more likely? At least 51 percent of the time, the researchers claim, and possibly as much as 55 percent to 60 percent -- depending on the flipping motion of the individual.In other words, more than random luck is at work."
I've put this post on the Other page for future reference. If you teach probability in school this is a good topic to "go beyond" the basic curriculum. Too many students learn that the probability of flipping heads is 1/2 and not that the probability of flipping heads on a fair coin is 1/2. And the coins around us in the real world aren't fair.
Here are some stories that caught my eye this week:
- What do you do with your spare time? According to Sott.net, "A 15-year-old boy believes he has discovered a forgotten Mayan city using satellite photos and Mayan astronomy. William Gadoury, from Quebec, came up with the theory that the Maya civilization chose the location of its towns and cities according to its star constellations. He found Mayan cities lined up exactly with stars in the civilization's major constellations. Studying the star map further, he discovered one city was missing from a constellation of three stars. Using satellite images provided by the Canadian Space Agency and then mapped on to Google Earth, he discovered the city where the third star of the constellation suggested it would be."
- ZeroHege has an article showing just how deep the economic problems bite, "According to Bloomberg, a new survey by Discover Financial Services found that 48% of parents think their child should pay a portion (if not all) of the cost of attending college, up from 39% four years ago. And just how will potential students pay that portion? Why, student loans of course. 32% of respondents said they would ask the bank for help, while 27% plan to rely on family savings, 4% said they would dip into retirement funds, and 3% even indicated that they may refinance their home to pay for their kids college."
- USNews has an excellent article on just how badly our schools are letting us down, and there is an economic price. "One in four who enter college immediately after high school graduation must pay college-level prices for high school-level classes....But before you write off inadequate high school preparation as a function of a student's family background or the type of college they attend, know this: Nearly half of first-year remedial students come from middle-class, upper middle-class and wealthy families. Forty percent are enrolled at public and private four-year colleges...We already knew that high schools typically underserve students from low-income families and communities, but apparently they're doing poorly with wealthier students as well. It turns out that all students are susceptible to the leaky K-12-to-college pipeline – no one is immune. This should be a wake-up call for all."
- Say what? CBS6albany starts an article with, ""Stop for a second and think about the standards that you graduated high school with, it's night and day," said Shenendehowa Central School District Superintendent Dr. Oliver Robinson." and I'm thinking, right! Keep reading some more, though, "Under current state regulations, and in the age of Common Core, there's a real concern among parents about their kids making it to the stage in cap and gown to receive a high school diploma - and it's only getting tougher." Wait, what? The educational system has gotten weaker over the years. Try finding someone under, say, 30 to 35 who can handle decimals, fractions and percents. Whereas taking calculus used to be rare, nowadays "good" math students are taking calculus--and they can't handle fractions. "Parents and educators across the state are expressing worries about the impact on graduation rates. Robinson says data predicts a severe drop."We would go from about 93% graduation rates to barely over 50%, significant," he said...."We'd have a situation where, to be quite frank, districts would not produce graduates," he explained." And THAT is one of the many reasons why the public school system is so messed up. High school is something you have to go through before you go on to a college, enjoy parties, go into debt and come out with a degree that you need to get a job because everybody has a worthless high school degree. Rather than uphold a standard that students who graduate learn math, grammar, spelling, history, etc so that students who graduate are college ready we now have a system where 37% are college ready and "Only 8 percent of U.S. high school graduates complete a curriculum that prepares them well for college and the workplace" all while "The nation's high school graduation rate hit 82 percent in 2013-14, the highest level since states adopted a new uniform way of calculating graduation rates five years ago.". When you make something free, it loses its value. Make students earn a high school degree and some might take it more seriously. College has become the "high school" of 50 years ago except you go into debt and there's still no guarantee you can do basic math when you're finished.