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.

 

The grounding for many further concepts in business, as in other applied disciplines, is a thorough understanding of statistics. A good free source for understanding probability is Introduction to Probability. In particular regression analysis is going to be useful. Regression is fundamental to the concept of Beta in Finance and conjoint analysis in Marketing. In addition, the statistical concepts of standard deviation and variance are the fundamental concepts which underly the principle of pooling risk in operations management.
The directions on how to perform a regression analysis in Excel can be found online at Regression Analysis Using Microsoft Excel. Once you have completed a few regressions in Excel, you have performed the highest level of statistical analysis which most MBA's become familiar with. Congratulations! You are ready to move on.