How often is battery saver on?

Question

A phone's battery-saver mode is either on or off each hour, and the next hour depends only on this one:

If on now, it turns off next hour with probability $0.3$ (and stays on with probability $0.7$ ).
If off now, it turns on next hour with probability $0.1$ (and stays off with probability $0.9$ ).

In the long run, what fraction of hours is battery saver on?

Accepted Answer

We want the steady shares (\pi{	ext{on}}, \pi{	ext{off}}) that stop changing hour to hour. Balance the flow out of "on" against the flow in. Flow leaving on: \pi{	ext{on}} 	imes 0.3. Flow entering on: \pi{	ext{off}} 	imes 0.1. Balance: \pi{	ext{on}} 	imes 0.3 = \pi{	ext{off}} 	imes 0.1 \implies \pi{	ext{on}} = 	frac13\,\pi{	ext{off}}. Since the shares add to 1: 	frac13\pi{	ext{off}} + \pi{	ext{off}} = 1 \implies 	frac43\,\pi{	ext{off}} = 1 \implies \pi{	ext{off}} = 	frac34, \quad \pi{	ext{on}} = 	frac14. So battery saver is on one quarter of the time, or 25\%. Two-state …

How often is battery saver on?

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