Around a ring of nine, how many top both neighbors?

Question

Nine sensors, each reporting a distinct reading, are arranged in a completely random order around a circular ring of 9 slots. Call a sensor a "hotspot" if its reading is higher than both of its two neighboring sensors.

On average, how many hotspots are there around the ring?

Accepted Answer

Put one indicator on each slot. On a circle every slot has exactly 2 neighbors, so all slots behave the same way. Fix a sensor. Consider the 3 distinct readings of that sensor and its 2 neighbors. All orderings of these 3 values are equally likely, and the sensor is a hotspot exactly when its reading is the largest, which happens in 	frac13 of orderings: P(	ext{a given sensor is a hotspot}) = \frac13. By linearity over all 9 sensors, E[	ext{hotspots}] = 9 	imes \frac13 = 3. Takeaway. On a ring, uniformity across slots makes the count clean: every slot is a local max with probability 	fra…

Around a ring of nine, how many top both neighbors?

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