Posts Tagged probability
Making random decisions on the basis of a coin flip
Posted by mark in Basic stats & math on August 28th, 2009
I watched the movie Leatherheads last night – a comedic tribute to the early days of professional football. It re-created the first coin flip used to determine which team would kick off. The referee allowed the coin to fall to the ground – introducing a bit more randomness into the outcome (as opposed him catching it). That’s one of the findings presented in the Dynamical Bias in the Coin Toss by a trio of mathematics and statistics professors from Stanford and UC Santa Cruz. Surprisingly, they report that for “natural flips” the chance of a coin coming up as started is 51 percent. In other words, this procedure for creating an even probability is biased by the physics.
My “heads up” (ha ha) on this came from a former colleague of mine. He sent me a link that led me to this flippant (pun-intended) summary by blogger James Devlin . Devlin warns against spinning a coin to create a 50/50 outcome – a heavy-headed coin can fall tails-up as much as 80 percent of the time! It seems to me that this approach would also increase the odds of a flipistic singularity – normally very rare (1 in 6000 chance).
Another colleague, who once collected comics about Donald Duck, told me the tale of flipism – a random way to live life. However, I think I will not go down this road, but rather quit this blog while I am still ahead.
“Life is but a gamble! Let Flipism chart your ramble.”
- Slogan in Flip Decision by Carl Barks
PS. A fellow trainer starts off statistics workshops with a fun icebreaker that gets students involved with flipping a coin. He asks the class what they expect for an outcome and then challenges this assumption experimentally. The first student gets heads, which the trainer tallies on a flipchart. Each student in turn gets the same outcome until someone finally gets suspicious and discovers that it’s a two-headed quarter.
Overreacting to patterns generated at random – Part 2
Professor Gary Oehlert provided this heads-up as a postscript on this topic:
“You might want to look at Diaconsis, Persi, and Fredrick Moesteller, 1989, “Methods for Studying Coincidences” in the Journal of the American Statistical Association, 84:853-61. If you don’t already know, Persi was a professional magician for years before he went back to school (he ran away from the circus to go to school). He is now at Stanford, but he was at Harvard for several years before that.”
I found an interesting writeup on Percy Diaconis and a bedazzling photo of him at Wikipedia. The article by him and Moesteller notes that “Coincidences abound in everyday life. They delight, confound, and amaze us. They are disturbing and annoying. Coincidences can point to new discoveries. They can alter the course of our lives; where we work and at what, whom we live with, and other basic features of daily existence often seem to rest on coincidence.”
However, they conclude that “Once we set aside coincidences having apparent causes, four principles account for large numbers of remaining coincidences: hidden cause; psychology, including memory and perception; multiplicity of endpoints, including the counting of “close” or nearly alike events as if they were identical; and the law of truly large numbers, which says that when enormous numbers of events and people and their interactions cumulate over time, almost any outrageous event is bound to occur. These sources account for much of the force of synchronicity.”
I agree with this skeptical point of view as evidenced by my writing in the May 2004 edition of the Stat-Ease “DOE FAQ Alert” on Littlewood’s Law of Miracles, which prompted Freeman Dyson to say “The paradoxical feature of the laws of probability is that they make unlikely events happen unexpectedly often.“
Overreacting to patterns generated at random — Part 1
My colleague Pat Whitcomb passed along the book Freakonomics to me earlier this month. I read a story there about how Steven D. Levitt, the U Chicago economist featured by the book, used statistical analysis To Catch a Cheat –teachers who improved their students’ answers on a multiple-choice skills assessment (Iowa Test). The book provides evidence in the form of an obvious repeating of certain segments in otherwise apparently-random answer patterns from presumably clueless students.
Coincidentally, the next morning after I read this, Pat told me he discovered a ‘mistake’ in our DX7 user guide by not displaying subplot factor C (Temp) in random run order. The data are on page 12 of this Design-Expert software tutorial on design and analysis of split plots. They begin with 275, 250, 200, 225 and 275, 250, 200, 225 in the first two groupings. Four out the remaining six grouping start with 275. Therefore, at first glance of this number series, I could not disagree with Pat’s contention, but upon further inspection it became clear that the numbers are not orderly. On the other hand, are they truly random? I thought not. My hunch was that the original experimenter simply ordered numbers arbitrarily rather than using a random number generator.*
I asked Stat-Ease advisor Gary Oehlert. He says “There are 4 levels, so 4!=12 possible orders. You have done the random ordering 9 times. From these 9 you have 7 unique ones; two orders are repeated twice. The probability of no repeats is 12!/(3!*12^12). This equates to a less than .00001 probability value. Seven unique patterns, as seen in your case, is about the median number of unique orders.”
Of course, I accept Professor Oehlert’s advice that I should not concern myself with the patterns exhibited in our suspect data. One wonders how much time would be saved by mankind as a whole by worrying less over what really are chance occurrences.
*The National Institute of Standards and Technology (NIST) provides comprehensive guidelines on random number generation and testing– a vital aspect of cryptographic applications.