Actually, we don’t know that our decision affects the contents of Box B. In fact, we’re told that it contains a million dollars if-and-only-if Omega predicts we will only take Box B.
It is possible that we could pick Box B even tho Omega predicted we would take both boxes. Omega has only observed to have predicted correctly 100 times. And if we are sufficiently doubtful whether Omega would predict that we would take only Box B, it would be rational to take both boxes.
Only if we’re somewhat confident of Omega’s prediction can we confidently one-box and rationally expect it to contain a million dollars.
Actually, we don’t know that our decision affects the contents of Box B. In fact, we’re told that it contains a million dollars if-and-only-if Omega predicts we will only take Box B.
It is possible that we could pick Box B even tho Omega predicted we would take both boxes. Omega has only observed to have predicted correctly 100 times. And if we are sufficiently doubtful whether Omega would predict that we would take only Box B, it would be rational to take both boxes.
Only if we’re somewhat confident of Omega’s prediction can we confidently one-box and rationally expect it to contain a million dollars.
51% confidence would suffice.
Two-box expected value: 0.51 $1K + 0.49 $1.001M = $491000
One-box expected value: 0.51 $1M + 0.49 $0 = $510000