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psychic plays the game all year round, never raises the stakes, and always quits at a profit of a million dollars, the expected return is $177,353,500.5

Clearly, Bem’s psychic could bankrupt all casinos on the planet before anybody real- ized what was going on. This analysis leaves us with two possibilities. The first possibility is that, for whatever reason, the psi effects are not operative in casinos, but they are op- erative in psychological experiments on erotic pictures. The second possibility is that the psi effects are either nonexistent, or else so small that they cannot overcome the house advantage. Note that in the latter case, all of Bem’s experiments overestimate the effect.

Returning to Laplace’s Principle, we feel that the above reasons motivate us to assign our prior belief in precognition a number very close to zero. For illustrative purposes, let us s e t P ( H 1 ) = 1 0 2 0 , t h a t i s , . 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 . T h i s m e a n s t h a t P ( H 0 ) = 1 P ( H .99999999999999999999. Our aim here is not to quantify precisely our personal prior belief in psi. Instead, our aim is to explain Laplace’s Principle by using a concrete example and specific numbers. It is also important to note that the Bayesian t-test outlined in the next 1 ) = s e c t i o n d o e s n o t d e p e n d i n a n y w a y o n t h e p r i o r p r o b a b i l i t i e s P ( H 0 ) a n d P ( H 1 ) .

Now assume we find a flawless, well-designed, 100% confirmatory experiment for w h i c h t h e o b s e r v e d d a t a a r e u n l i k e l y u n d e r H 0 b u t l i k e l y u n d e r H 1 , s a y b y a f a c t o r o f 1 9 ( indicated below, this is considered “strong evidence”). In order to update our prior belief, we apply Bayes’ rule: a s

p ( H 1 | D ) =

p ( D | H 1 ) p ( H 1 ) p ( D | H 0 ) p ( H 0 ) + p ( D | H 1 ) p ( H 1 )

.95 × 1020 = .05(1 1020) + .95 × 10


= .00000000000000000019.

True, our posterior belief in precognition is now higher than our prior belief. Nevertheless, we are still relatively certain that precognition does not exist. In order to overcome our skeptical prior opinion, the evidence needs to be much stronger. In other words, extraor- dinary claims require extraordinary evidence. This is neither irrational nor unfair; if the proponents of precognition succeed in establishing its presence, their reward is eternal fame, (and, if Bem were to take his participants to the casino, infinite wealth).

Thus, in order to convince scientific critics of an extravagant or controversial claim, one is required to pull out all the stops. Even when Bem’s experiments had been confir- matory (which they were not, see above), and even if they would have conveyed strong statistical evidence for precognition (which they did not, see below), eight experiments are not enough to convince a skeptic that the known laws of nature have been bent. Or, more precisely, that these laws were bent only for erotic pictures, and only for participants who are extraverts.

5The break-even point for the house lies at a success probability of 0.514. However, even if the success rate is smaller, say, 0.510, one can boost one’s success probability by utilizing a team of psychics and using their majority vote. This is so because Condorcet’s jury theorem ensures that, whenever the success probability for an individual voter lies above 0.5, the probability of a correct majority vote approaches 1 as the number of voters grows large. If the individual success probability is 0.510, for instance, using the majority vote of a team of 1000 psychics gives a probability of .73 for the majority vote being correct.

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