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Introduction to Probability and Statistics
Calculation 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.
But suppose this subject continues to guess about 60 right out of a hundred, so that after ten runs of 100 tosses—1000 tosses in all, the subject has made 600 correct guesses. The probability of that happening purely by chance is less than one in seven billion, so it's time to start thinking about explanations other than luck. Still, improbable things happen all the time: if you hit a golf ball, the odds it will land on a given blade of grass are millions to one, yet (unless it ends up in the lake or a sand trap) it is certain to land on some blades of grass.
Finally, suppose this “dream subject” continues to guess 60% of the flips correctly, observed by multiple video cameras, under conditions prescribed by skeptics and debunkers, using a coin provided and flipped by The Amazing Randi himself, with a final tally of 1200 correct guesses in 2000 flips. You'd have to try the 2000 flips more than 5×1018 times before you'd expect that result to occur by chance. If it takes a day to do 2000 guesses and coin flips, it would take more than 1.3×1016 years of 2000 flips per day before you'd expect to see 1200 correct guesses due to chance. That's more than a million times the age of the universe, so you'd better get started soon!
Claims of evidence for the paranormal are usually based upon statistics which diverge so far from the expectation due to chance that some other mechanism seems necessary to explain the experimental results. To interpret the results of our RetroPsychoKinesis experiments, we'll be using the mathematics of probability and statistics, so it's worth spending some time explaining how we go about quantifying the consequences of chance.
Note to mathematicians: The following discussion of probability is deliberately simplified to consider only binomial and normal distributions with a probability of 0.5, the presumed probability of success in the experiments in question. I decided that presenting and discussing the equations for arbitrary probability would only decrease the probability that readers would persevere and arrive at an understanding of the fundamentals of probability theory.
Twelve and a half cents: one bit!
In slang harking back to the days of gold doubloons and pieces of eight, the United States quarter-dollar coin is nicknamed “two bits”. The Fourmilab radioactive random number generator produces a stream of binary ones and zeroes, or bits. Since we expect the generator to produce ones and zeroes with equal probability, each bit from the generator is equivalent to a coin flip: heads for one and tails for zero. When we run experiments with the generator, in effect, we're flipping a binary coin, one bit—twelve and a half cents!
(We could, of course, have called zero heads and one tails; since both occur with equal probability, the choice is arbitrary.) Each bit produced by the random number generator is a flip of our one-bit coin. Now the key thing to keep in mind about a genuine random number generator or flip of a fair coin is that it has no memory or, as mathematicians say, each bit from the generator or flip is independent. Even if, by chance, the coin has come up heads ten times in a row, the probability of getting heads or tails on the next flip is precisely equal. Gamblers who've seen a coin come up heads ten times in a row may believe “tails is way overdue”, but the coin doesn't know and couldn't care less about the last ten flips; the next flip is just as likely to be the eleventh head in a row as the tail that breaks the streak.
Even though there is no way whatsoever to predict the outcome of the next flip, if we flip a coin a number of times, the laws of probability allow us to predict, with greater accuracy as the number of flips increases, the probability of obtaining various results. In the discussion that follows, we'll ignore the order of the flips and only count how many times the coin came up heads. Since heads is one and tails is zero, we can just add up the results from the flips, or the bits from the random generator.
Four Flips
Suppose we flip a coin four times. Since each flip can come up heads or tails, there are 16 possible outcomes, tabulated below, grouped by the number of heads in the four flips.
Number of Ways summarises how many different ways the results of the four flips could end up with a given number of heads. Since the only way to get zero heads is for all four flips to be tails, there's only one way that can occur. One head out of four flips can happen four different ways since each of the four flips could have been the head. Two heads out of four flips can happen six different ways, as tabulated. And since what's true of heads applies equally to tails, there are four ways to get three heads and one way to get four.
Mathematically, the number of ways to get x heads (or tails) in n flips is spoken of as the “number of combinations of n things taken x at a time”, which is written as:
c 
This, it turns out, can be calculated for any positive integers n and x whatsoever, as follows.
For example, if we plug in 4 for n and 2 for x, we get
4! / (2! (4 − 2)!) = 4! / (2! 2!) = 24 / (2 2) = 24 / 4 = 6
as expected. Plotting the number of ways we can get different numbers of heads yields the following graph.
Probability
Since the coin is fair, each flip has an equal chance of coming up heads or tails, so all 16 possible outcomes tabulated above are equally probable. But since there are 6 ways to get 2 heads, in four flips the probability of two heads is greater than that of any other result. We express probability as a number between 0 and 1. A probability of zero is a result which cannot ever occur: the probability of getting five heads in four flips is zero. A probability of one represents certainty: if you flip a coin, the probability you'll get heads or tails is one (assuming it can't land on the rim, fall into a black hole, or some such).
The probability of getting a given number of heads from four flips is, then, simply the number of ways that number of heads can occur, divided by the number of total results of four flips, 16. We can then tabulate the probabilities as follows.
Since we are absolutely certain the number of heads we get in four flips is going to be between zero and four, the probabilities of the different numbers of heads should add up to 1. Summing the probabilities in the table confirms this. Further, we can calculate the probability of any collection of results by adding the individual probabilities of each. Suppose we'd like to know the probability of getting fewer than three heads from four flips. There are three ways this can happen: zero, one, or two heads. The probability of fewer than three, then, is the sum of the probabilities of these results, 1/16 + 4/16 + 6/16 = 11/16 = 0.6875, or a little more than two out of three. So to calculate the probability of one outcome or another, sum the probabilities.
To get probability of one result and another from two separate experiments, multiply the individual probabilities. The probability of getting one head in four flips is 4/16 = 1/4 = 0.25. What's the probability of getting one head in each of two successive sets of four flips? Well, it's just 1/4 × 1/4 = 1/16 = 0.0625.
The probability for any number of heads x in any number of flips n is thus:
the number of ways in which x heads can occur in n flips, divided by the number of different possible results of the series of flips, measured by number of heads. But there's no need to sum the combinations in the denominator, since the number of possible results is simply two raised to the power of the number of flips. So, we can simplify the expression for the probability to:
More Flips
Let's see how the probability behaves as we make more and more flips. Since we have a general formula for calculating the probability for any number of heads in any number of flips, we can graph of the probability for various numbers of flips.
8 Flips 
16 Flips 
32 Flips 
64 Flips 
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