Expected number of sensors that trigger

Question

A room has 6 motion sensors. Each one independently triggers with probability 0.3 during a test. On average, how many of the 6 sensors trigger?

Accepted Answer

Put one indicator on each sensor: Xi = 1 if sensor i triggers. Each has E[Xi] = 0.3. The number of triggers is X = X1 + \cdots + X6, and by linearity of expectation E[X] = 6 	imes 0.3 = 1.8. When all the probabilities are equal, the sum is just the count times the shared probability, which is the mean of a binomial distribution, np. Takeaway. With equal probabilities the expected count is np: here 6 	imes 0.3 = 1.8 sensors on average.

Expected number of sensors that trigger

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