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.
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.