Original practice · fully worked
Original variant: expected incident count after an audit pass
Half of service-desk shifts have zero incidents, three tenths have one, and the remainder have three. An audit passes in 90%, 60%, and 20% of those respective states. Find the expected incident count given that the audit passes.
- C
0.2687 - B
0.3750 - A
0.4478 - D
0.6000 - E
0.7463
Answer and calculation
The pass-state weights are 0.45, 0.18, and 0.04. Their total is 0.67 and their count-weighted sum is 0.30, giving conditional mean 0.4478.
Setup
Setup
Combine each incident-count probability with its audit-pass probability to form the three joint pass weights.
P(N=0,1,3)=(0.50,0.30,0.20) Model
Model
The joint weights for counts 0, 1, and 3 are 0.45, 0.18, and 0.04.
P(A∣N=0,1,3)=(0.90,0.60,0.20) Compute
Compute
The pass probability is 0.67 and the count-weighted pass total is 0.30, giving conditional mean 0.447761.
P(A)=0.45+0.18+0.04=0.67 E[N1A]=1(0.18)+3(0.04)=0.30 E[N∣A]=0.30/0.67=0.447761 Answer
Answer
The expected incident count given a pass is 0.4478, corresponding to choice A.
0.4478(A) 