The following TED video, given by mathemagician and professor Arthur Benjamin (about whom I’ve previously blogged about here), embodies the best idea I’ve heard about math education in a LONG time. Perhaps ever. Just as I recently posted about how games like backgammon embody the 21st century in replacement of games like chess for the 20th, statistics is the central branch of mathematics for the 21st century rather than the calculus centric view of the 20th century. If you’re into math and math education, this will probably be the best 3 minutes you’ll spend today.
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Dan Gilbert has a few talks posted on the TED website, and the one that’s embedded here has to do with the idea of expected value, and how this mathematical idea applies to practical life. The idea of expected value was formalized by Bernoulli, and Dan explains it as the product of “the odds that an action will allow us to gain something, and the value of the gain to us”. Quite simply, an expected values is a way of quantifying whether or not a decision is a good one. If you watch the first 5 minutes of the video below, there are some simple examples given, but I’ll throw one out as well.
Imagine that I walk up to you, and I say the following: “I’ll tell you what. Right now, I’m thinking about a color that’s in a rainbow. If you can guess what color I’m thinking about, I’ll give you $14. If you guess incorrectly, you give me $1.” Do you take the bet?
Well, assuming that I’m not cheating in any way, and there are 7 colors in a rainbow (ROYGBIV), you have a 1/7 chance of guessing correctly. And if you win, you get $14. Given the definition from above, the expected value of this bet is equal to (1/7) * ($14) = $2. In other words, given this betting situation you stand to make $2. And our arrangement dictates that if I win, you give me $1. Comparing your possible dollar gains or losses, this is certainly a good bet. You should take it.
Where this idea gets extremely interesting is when it’s applied to human psychology and everyday decision making. Basically, we’re very bad at computing expected values in our daily lives. Obviously, I don’t expect that any of you generally has someone coming up to you offering bets of the aforementioned type, but decisions bombard us daily.
The reason I enjoy this talk so much is that it takes a mathematical idea and tells us why it’s PRACTICAL. Gasp! There are a lot of interesting examples given in this talk, and Dan is an engaging speaker. I certainly recommend this video as a worthwhile way to spend half an hour!
If you haven’t yet been introduced to the Stuff White People Like blog, you’re in for a treat. There is a new article on statistics that you can read here. Basically, the blog is comprised of witty, yet strangely accurate descriptions of…well…things that white people like. It’s truly hilarious. For instance, take the first line of this article, “White people hate math. If you want to befriend white people, mention “that weird Asian calculus teacher who drew perfect circles” and how much you hated his class…” Awesome.
A few days ago on Slashdot there was an article about a statistical model that claims to be able to accurately predict the result of a war nearly 4 out of 5 times. Here’s a snippet from the University of Georgia’s press release on Dr. Patricia L. Sullivan’s study: “‘If you know some key variables – like the major objective, the nature of the target, whether there’s going to be another strong state that will intervene on the side of the target and whether you’ll have an ally – you can get a sense of your probability of victory,’ said Sullivan, whose study appears in the June issue of the Journal of Conflict Resolution.” Very interesting. Statistics is a beautiful, and very misunderstood, field. When I hear about claims like this my ears definitely perk up. In general, studies like this propose that particular variables (such as a poor military strategy) are predictive of other events (such as a military victory). There’s obviously a cause/effect chain reflected in this type of idea. And believe it or not, there is a LOT of study in the area of cause/effect relationships. People like Peter Spirtes at Carnegie Mellon University spend a lot of time studying these causal relationships.
So while that claim that a statistical model can predict the outcome of wars should be taken with a grain of salt, everyone should consider the fantastic amount of research (and quality science) that is going into these types of causal models.
As if we didn’t already have enough to thank Guinness for, I learned something very interesting today about a connection between this fine beer and statistics. I’m reading the book “Randomness” by Deborah Bennett (Amazon link), which is an introductory text concerning the basics of probability theory and statistics. It’s an accessible read, and contains several nuggets of interesting historical information. I suggest checking it out if you’re interested in this sort of thing. Here’s what Dr. Bennett has to say concerning the Guinness connection to statistics:
“The best-known early demonstration of a random sampling experiment was performed by William Sealy Gosset, a research chemist working for the Arthur Guinness Son and Company Ltd. in Dublin. Gosset was studying the relationship between the quality of Guinness beer and various factors in the beer’s production. The brewery was continually experimenting with soil conditions and grain variety that might produce improvements in crop yield, and Gosset was intent on bringing all the benefits of statistics to the brewery’s agricultural experiments.”
In short, Gosset discovered the t-distribution while working on this problem for Guinness. If you’ve ever taken an introductory course in statistics you’ll probably remember working with t-tests. They’re a way of correctly analyzing small sample sizes, where “small” usually means samples less than size 30. There are some other fun facets of Gosset’s work on this problem, including the fact that he published his findings in papers under the pseudonym of Student. Read Gosset’s Wikipedia page here for other general pieces of information.
I don’t know what the hell this article seems to be talking about. I think that the article would like me to believe that several professors at a school in the UK have somehow “cracked” the lottery system. Well, let’s examine their method. They “bagged the big prize by using two boxes, 49 pieces of paper and a large amount of brain power”. Check. It took the group 4 years and about $9000, but they actually hit the six digit number to the tune of $13 million.
Ahem…
Luck. These people hit the lottery. These people didn’t “crack” anything. This so-called system of “brain power”: luck. Though I don’t know all of the particulars regarding the lottery game they won, I can be sure that their method didn’t vastly improve their odds. (If anyone knows exactly how the odds were changed using the method described in the article please let me know.)
It’s PROBABILITY, people! The state wouldn’t run a lottery if they didn’t make money off of it. The lottery is designed to make the state money. And I assure you that they make a LOT of money off of us playing.
So please, don’t let articles like this fool you. There is no system for winning the lottery. At least not one that’s profitable. Or legal. So go invest your money in a house or a mutual fund. Consider this my public service announcement of the day.
There’s an interesting article over at ABC News explaining that the number of Americans buying into evolutionary theory has been in decline. The story attributes this surge of opinion to several sources, including “religious fundamentalism, inadequate science education, and partisan political maneuvering”. While I’m not currently interested in discussing the relevancy of these possibilities, the article also talks about the misuse of probability theory by creationists. Here’s a snippet of the argument:
A bit more specifically, the standard argument goes roughly as follows. A very long sequence of individually improbable mutations must occur in order for a species or a biological process to evolve. If we assume these are independent events, then the probability of all of them occurring and occurring in the right order is the product of their respective probabilities, which is always an extremely tiny number. Thus, for example, the probability of getting a 3, 2, 6, 2, and 5 when rolling a single die five times is 1/6 x 1/6 x 1/6 x 1/6 x 1/6 or 1/7,776 — one chance in 7,776.
Check out the above link to read the rest of the story.
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