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Nobody has brought up the [Conjunction Fallacy](http://en.wikipedia.org/wiki/Conjunction_fallacy)--basically a cognitive bias in humans that sometimes makes people think that "specific conditions are more probable than a single general one."

Example from Wikipedia:

&gt;

    Linda is 31 years old, single, outspoken, and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in anti-nuclear demonstrations.

    Which is more probable?

        Linda is a bank teller.
        Linda is a bank teller and is active in the feminist movement.

People tend to say that the second scenario is more likely, which is impossible, because in any scenario where the second is true, the first is necessarily true. 

The problem with the report is a slightly different but related version of the fallacy, which might be illustrated by the following example:

    Which is more probable?

        Linda is a bank teller and is active in the feminist movement.
        Linda is not active in the feminist movement.

So even though each *individual* part of the first claim may seem more likely than the second claim, the combined probability of the first claim may be lower: After all, bank tellers make up a very small percentage of the population, and it's not implausible that somebody with feminist ideals may have stopped being "active" in the movement (or even changed their ideals).  So even though "Linda is a bank teller" and "Linda is active in the feminist movement" seem more likely than "Linda is not active in the feminist movement," *combined* they may be less likely.

This transfers to the Brady situation in the following way:

    It is more likely than not that the Patriots tampered with the balls, and IF they tampered with the balls, it is more likely than not that Tom Brady knew about and authorized the tampering.

    Which is more probable?

        The Patriots tampered with the balls AND Brady knew about it.
        Brady did not know about any ball tampering.

The issue combines that when you find the probability of a conjunction, you have to MULTIPLY the probabilities of the individual statements: For example, say that they are 75% sure that the Patriots tampered, and 60% that if they did tamper then Brady knew about it, then the probability that Brady knew about any tampering is .75\*.6 = 45%, which would mean it is more likely than not that Brady did not know about it. Of course these probabilities were made up arbitrarily, but they illustrate the central issue: **Even if it is more probable than not that the Patriots tampered, and more probable than not that if they did tamper then Brady knew about it, it is still possible for it to be more probable than not that Brady did not know about any tampering.**
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