Saturday Night Science: Introduction to Probability and Statistics

DiceCalculation and Chance

Most experimental searches for paranormal phenomena are statistical in nature. A subject repeatedly attempts a task with a known probability of success due to chance, then the number of actual successes is compared to the chance expectation. If a subject scores consistently higher or lower than the chance expectation after a large number of attempts, one can calculate the probability of such a score due purely to chance, and then argue, if the chance probability is sufficiently small, that the results are evidence for the existence of some mechanism (precognition, telepathy, psychokinesis, cheating, etc.) which allowed the subject to perform better than chance would seem to permit.

Suppose you ask a subject to guess, before it is flipped, whether a coin will land with heads or tails up. Assuming the coin is fair (has the same probability of heads and tails), the chance of guessing correctly is 50%, so you’d expect half the guesses to be correct and half to be wrong. So, if we ask the subject to guess heads or tails for each of 100 coin flips, we’d expect about 50 of the guesses to be correct. Suppose a new subject walks into the lab and manages to guess heads or tails correctly for 60 out of 100 tosses. Evidence of precognition, or perhaps the subject’s possessing a telekinetic power which causes the coin to land with the guessed face up? Well,…no. In all likelihood, we’ve observed nothing more than good luck. The probability of 60 correct guesses out of 100 is about 2.8%, which means that if we do a large number of experiments flipping 100 coins, about every 35 experiments we can expect a score of 60 or better, purely due to chance.... [Read More]

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TOTD 2018-04-28: “The Surprisingly Solid Mathematical Case of the Tin Foil Hat Gun Prepper”

Here is an article you should read and think about: “The Surprisingly Solid Mathematical Case of the Tin Foil Hat Gun Prepper”.  The author is a stormwater hydrologist—what he does for a living is study the behaviour of water as it moves through the Earth’s ecosystem and, in particular, extreme events such as floods.  It is he and his colleagues who draw the red lines on maps which determine whether you can get flood insurance at an affordable price and, in many cases, obtain a long-term mortgage on a property.  Those in his profession think deeply about “tail risks”: events which occur rarely but which have major, or even catastrophic, consequences when they happen.  Humans have evolved in an environment which has selected them to apply a number of heuristics that, in most cases, get a good enough answer without a complete understanding of the problem or an exhaustive analysis of the situation.  But evolution, biological or cultural, is poor at selecting for heuristics which apply for events which happen less frequently than the lifetime of most members of a species.  We use our intuition, and often we get the wrong answer.

There is some math in the cited paper, and in what follows, but nothing more complicated than multiplying numbers, which you can do with a calculator if you wish to work it out for yourself.  The only other thing you need to know are some very basic facts of probability and statistics: I’ve written an “Introduction to Probability and Statistics” which, while aimed at other applications, may help if you’re rusty and want to review the details.  All you really need to know is that if a series of events are independent of one another (the outcome of one doesn’t affect the others), and that they have a given individual probability, you get the probability of a series of events occurring one after another by multiplying their probabilities together.... [Read More]

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