Random Switch Threshold - Quant Problems

Approach

Let K be the number of switches that are on. Conditional on K = k, the event succeeds exactly when C is one of 0,1,...,k, so its conditional probability is (k+1)/9. Therefore the desired probability is E[(K+1)/9]. By linearity of expectation, E[K] is the sum of the eight switch probabilities, which is 39/10. The probability is (39/10+1)/9 = 49/90.

Approach

Let K be the number of switches that are on. Conditional on K = k, the event succeeds exactly when C is one of 0,1,...,k, so its conditional probability is (k+1)/9. Therefore the desired probability is E[(K+1)/9]. By linearity of expectation, E[K] is the sum of the eight switch probabilities, which is 39/10. The probability is (39/10+1)/9 = 49/90.