Worked solution · Exam P
This Exam P sample reference tests Expected Value. Every covered employee selects exactly two items, so the sum of the three marginals is twice the selection probability. The remaining probability is 1/2, choice C.
Independent solution
Setup
Let q be the probability that one of the possible item pairs is selected.
Model
Whenever a pair is selected, exactly two marginal item indicators equal one. Therefore the sum of the three marginal probabilities is 2q.
Compute
The marginal probabilities sum to one, so 2q=1 and q=1/2.
Answer
The complementary probability that no pair is selected is also 1/2, which is choice C.
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Original practice · fully worked
Users either activate no dashboard widget, exactly one of widgets A, B, and C, or all three. The three activation marginals are 0.25, 0.30, and 0.35, and 0.08 of users activate all three. Find the probability that a user activates no widget.
The marginal sum counts each single activation once and each triple activation three times. Thus exactly-one mass is 0.66, total active mass is 0.74, and none is 0.26, choice A.
Setup
Let N be the number of activated widgets and add the three widget marginals to obtain the expected number activated.
Model
A one-widget user contributes one to the marginal sum and a three-widget user contributes three, so solve for the exactly-one mass.
Compute
The exactly-one mass is 0.90-3(0.08)=0.66. Subtracting the exactly-one and exactly-three masses from one leaves 0.26 for no widget.
Answer
Thus the no-widget probability is 0.26, corresponding to choice A.
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