Paradox Schmaradox

Warning: Spoiler below. Tem42 has pointed out, quite rightly, that it is more fun to figure out the paradox on your own, so try that first, if you're so inclined.

Principle 2 is your problem, here. (Referring to Tem42's above node.)

Suppose proposition *p* is 70% likely to be true, and proposition *q* is also 70% likely to be true. Each proposition, *taken individually*, is more likely to be true than not.

Now, imagine that you are walking to work, and since it rained last night, the road is a bit muddy. Suppose *p* represents whether or not you will make it to work without getting mud on your suit. And *q* represents whether or not a car will run through a puddle, splashing and soaking your suit.

Now, according to this Principle 2, one might assume, "Since it is probable that I will reach my workplace without getting mud on my suit, and since it is probable that I will get there without getting my suit wet, I can conclude that I will probably make it to work with a clean suit."

Of course, the probability that *p* AND *q* are true is not 70%, but the probabilities multiplied, or 49%. So, you will probably arrive at work with a wet or muddy suit, and should probably consider an alternate route.

So, even if it is 99.999% certain that an individual lottery ticket will lose, it is only ~99.998% certain that two tickets will **both** lose, and only ~99.997% certain that all of three tickets will lose, and so forth, with decreasing probabilities that **all** of an increasingly larger set of tickets will lose.

By the time you reach the 100,000th ticket, the probability that *all* will lose is some infinitesimal fraction very close to zero, which represents the probability that a large meteor will destroy the earth before the lottery is drawn.