It looks to me like there’s some confusion in the other comments regarding this. The expected value is, in fact, defined, and it is zero. The problem is that if you look at a sequence of n bets and take n to infinity, that expected value does go to positive infinity. So thinking in terms of adding one bet each time is actually deceiving.
In general, a sequence of pointwise converging random variables does not converge in expected value to the expected value of the limit variable. That requires uniform convergence.
Infinities sometimes break our intuitions. Luckily, our lives and the universe’s “life” are both finite.
The expected value is, in fact, defined, and it is zero.
Is the random variable you’re thinking of, whose expectation is zero, just the random variable that’s uniformly zero? That doesn’t seem to me to be the right way to describe the “bet” strategy; I would prefer to say the random variable is undefined. (But calling it zero certainly doesn’t seem to be a crazy convention.)
It’s zero on the event “three sixes are rolled at some point” and infinity on the event that they’re never rolled. The probability of that second event is zero, though. So the expected value is zero.
It looks to me like there’s some confusion in the other comments regarding this. The expected value is, in fact, defined, and it is zero. The problem is that if you look at a sequence of n bets and take n to infinity, that expected value does go to positive infinity. So thinking in terms of adding one bet each time is actually deceiving.
In general, a sequence of pointwise converging random variables does not converge in expected value to the expected value of the limit variable. That requires uniform convergence.
Infinities sometimes break our intuitions. Luckily, our lives and the universe’s “life” are both finite.
Is the random variable you’re thinking of, whose expectation is zero, just the random variable that’s uniformly zero? That doesn’t seem to me to be the right way to describe the “bet” strategy; I would prefer to say the random variable is undefined. (But calling it zero certainly doesn’t seem to be a crazy convention.)
It’s zero on the event “three sixes are rolled at some point” and infinity on the event that they’re never rolled. The probability of that second event is zero, though. So the expected value is zero.