This Exam P sample reference tests Law of Total Probability. Express the low- and high-group death probabilities as multiples of the medium-group probability, then apply the law of total probability. Solving the weighted equation gives approximately 0.009863, which corresponds to choice B.
How to solve this Law of Total Probability question
Setup
Setup
Let q denote the one-year death probability for a member of the medium group. Translate the two relative-risk statements into probabilities expressed in terms of q.
qM=q,qL=2q,qH=4q
Model
Model
The remaining population share is 0.15, so the overall probability is the weighted average of the three group probabilities.
0.15(2q)+0.55q+0.30(4q)=0.018
Compute
Compute
Collect the coefficients of q and solve the single linear equation.
(0.075+0.55+1.20)q=1.825q=0.018
q=1.8250.018=0.0098630137…
Answer
Answer
The medium-group probability rounds to the value displayed in choice B.
qM≈0.010(B)
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These notes identify the calculation error associated with each wrong letter when that error is reproducible.
AThis is the low-group probability, q_L=q_M/2=0.0049315..., rounded to 0.005; it answers for the wrong category.
CUsing a high-group multiplier of 2 instead of 4 gives q_M=0.018/1.225=0.014694..., which rounds to choice C.
DThis repeats the population-wide death probability 0.018 without resolving the mixture into its three group-specific rates.
EIf q_M were 0.021, the stated weights and multipliers would produce an overall probability of 1.825(0.021)=0.038325, not 0.018.
Original practice · fully worked
Original variant: field-sensor failure classes
A sensor fleet is divided among sheltered, standard, and harsh sites in proportions 0.20, 0.50, and 0.30. A standard-site sensor is three times as likely to fail during a year as a sheltered-site sensor, while a harsh-site sensor is twice as likely to fail as a standard-site sensor. The annual failure probability for a randomly selected sensor is 0.028. Calculate the annual failure probability at a standard site.
A 0.80%
B 1.40%
C 2.40%
D 2.80%
E 4.80%
Variant answer in brief
Write the sheltered and harsh failure rates as one-third and twice the standard-site rate. The weighted fleet equation gives a standard-site probability of 0.024, so the answer is C.
Setup
Setup
Let s be the standard-site annual failure probability and express the other two rates relative to it.
qsheltered=3s,qstandard=s,qharsh=2s
Model
Model
Average the three rates with the fleet proportions.
0.20(3s)+0.50s+0.30(2s)=0.028
Compute
Compute
The coefficient is 7/6, so the standard-site rate follows directly.
(151+21+53)s=67s=0.028
s=0.028(76)=0.024
Answer
Answer
The standard-site annual failure probability is listed in choice C.
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