Today on Salon there's a story which attempts to debunk the "psychic" John Edwards. Author Shari Waxman makes a good case of pointing out how Edwards manipulates the audience and works the odds in his favor, but nearly made me gag with the following:
But Edward, a 32-year-old native of Long Island, has not fessed up to all of his talents. As it happens, he is more than a psychic medium; he is also a master statistician. The smoke and mirrors behind his self-professed ability to communicate with the dead is a simple application of the summation law of probability. The law states that the calculated probabilities of events that are independent (i.e., the occurrence of one event has no effect on the probability that another event will occur) may be added together. In symbolic terms, where A is the first event, B is the second event and P stands for probability:
P(A) + P(B) = P(A or B)
For example, if you roll a six-sided die betting on a 3, your chances for success are 1 in 6, or 17 percent. Roll the die six times, and you are almost guaranteed to see a 3 (17 percent x 6 = 102 percent). Lucky for Edward, most audience members on his television show, "Crossing Over," are too hopeful and trusting to pull out a calculator and expose the charlatan behind the prophet.
However, Waxman is all wrong when she tries to extend this to successive events. If what she said were true, then the probability of seeing at least one heads on two flips of a coin would be 100%. And whoever proofread this piece should be shot, since a 102% probability is impossible.
The right way to figure out the probability of a single outcome A occurring over N tries is to calculate the probability of A not occurring at all, and then subtracting it from 100%. The odds of two events occurring together is the product of their probabilities. Thus, the odds of outcome A occurring on consecutive tries is P(A) x P(A), where P(A) means the probability of A as before.
Let's take Waxman's example of rolling a 3. The odds that you do not roll a 3 on a given toss of a six-sided die is 5/6. The odds of not rolling a 3 on consecutive tosses is the product of the probabilities, so for two tosses it's (5/6) x (5/6), or 25/36. For six tosses, it's (5/6) multiplied by itself six times (ie, to the sixth power), which works out to be about 33.4%. Since that's the odds of not rolling a 3, the odds of rolling at least one 3 is 66.6%, because the odds of an event occurring (in this case, no threes in six dice rolls) plus the odds of that same event not occurring (in this case, at least one three over six dice rolls) must add up to 100%.
Putting Waxman's mathematical foibles aside, I was happy with her debunking efforts until the very end:
I prefer to believe Edward's fans are not unintelligent, but simply in need of something to believe in, to feel good about, or to relieve the anxiety of what cannot be controlled. If he is fulfilling these needs, then in some ways, his gig is legit. Just like playing the lottery, if you really want to believe, you are better off not knowing the odds.