Smart lamp two steps ahead

Question

A smart lamp is either on or off each minute, controlled by a jumpy sensor whose next state depends only on the current one: - If the lamp is on now, it is on next minute with probability 0.6 (and off with probability 0.4). - If the lamp is off now, it is on next minute with probability 0.3 (and off with probability 0.7). Right now the lamp is off. What is the probability that the lamp is on two minutes from now?

Accepted Answer

Two minutes ahead means two steps, so we track what happens in the minute in between and keep only the routes that finish "on." We start off, the middle minute can be either state, and the final minute must be on. - Off, then off, then on: 0.7 	imes 0.3 = 0.21. - Off, then on, then on: 0.3 	imes 0.6 = 0.18. Add the two routes that end on: 0.21 + 0.18 = 0.39. So the lamp is on two minutes from now with probability 0.39. Takeaway. For a two-step question, branch on the intermediate state, multiply along each route, and sum the routes that reach your target. That is all a "two steps later" Mar…

Smart lamp two steps ahead

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