Spot the even split in a table

Question

A patrol drone is in one of three zones $X$ , $Y$ , $Z$ each hour. The next hour depends only on the current zone:

From $X$ : to $X$ with $0.4$ , to $Y$ with $0.1$ , to $Z$ with $0.5$ .
From $Y$ : to $X$ with $0.5$ , to $Y$ with $0.4$ , to $Z$ with $0.1$ .
From $Z$ : to $X$ with $0.1$ , to $Y$ with $0.5$ , to $Z$ with $0.4$ .

In the long run, what fraction of hours does the drone spend in zone $X$ ?

Accepted Answer

Before solving anything, check the columns, which collect all the ways to arrive at each zone: - Arriving at X: 0.4 + 0.5 + 0.1 = 1. - Arriving at Y: 0.1 + 0.4 + 0.5 = 1. - Arriving at Z: 0.5 + 0.1 + 0.4 = 1. Rows already sum to 1, and now every column sums to 1, so the chain is doubly stochastic. That guarantees the long-run distribution is even: an equal split stays equal, because the inflow to each zone is (equal share) 	imes 1. With three zones, \piX = \piY = \piZ = 	frac13. So the drone spends one third of its time in zone X, read straight off the structure without any equations. Takea…

Spot the even split in a table

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