Exam P technique guide
Learn how to recognize the method, recall the formulas that control it, and work through 55 sample-question solutions.
Try this original problem before opening the solution. Then compare the method with SOA Exam P sample question 10.
Original practice · fully worked
A data pipeline works only when each of three independent components works. Their operating probabilities during a run are 0.90, 0.80, and 0.75. Find the probability that the pipeline fails during a run.
All three components work with probability 0.90×0.80×0.75=0.54. The complementary failure probability is 0.46, choice E.
Setup
Let W be the event that all three components operate; the pipeline fails on the complementary event.
Model
Because the component states are independent, multiply their three operating probabilities to obtain Pr(W).
Compute
All components operate with probability 0.54, so at least one failure occurs with complementary probability 1-0.54=0.46.
Answer
The pipeline failure probability is 0.46, which is choice E.
Technique orientation
Independence is a property to verify, not a default shortcut. For events it means that learning one event does not alter the probability of the other; for random variables it means the joint law factors into its marginals. The required factorization should be written before products replace intersections.
Exam P deliberately contrasts independence with disjointness, exchangeability, and zero covariance. Questions also use independence to simplify repeated trials, maxima or minima, and variances of sums. The safe habit is to state the exact consequence needed and confirm that the hypothesis supports it.
Recognition audit
Before calculation
Name the requested quantity, mark its units or time point, and identify the assumption that permits the formula. Only then insert the numerical inputs. This separates model errors from arithmetic errors and makes the answer-choice comparison auditable.
Formula core
These identities are starting points, not substitutes for checking support, timing, rate periods, or independence assumptions.
Event independence
When the marginal probabilities are positive, this is equivalent to P(A | B) = P(A).
Independent random variables
For continuous variables the factorization must hold over the full joint support.
Variance consequence
The converse is generally false; zero covariance alone need not imply independence.
Three failure modes to catch
Verified practice ledger
The list is grouped by current official syllabus area. Every link opens a step-labeled solution, verification artifact, distractor analysis, and one fully worked original variant.
28 verified references
11 verified references
16 verified references
Continue the study loop
The complete Exam P manual includes all active sample-reference solutions, their wrong-choice analysis, and one independently written worked variant per reference. The free formula sheet is the compact companion for recall between practice sessions.
Related drills
Switch techniques when you can identify the current method before writing a formula and can explain each wrong-choice mechanism.
103 verified problems
Conditioning, total probability, and Bayes calculations organized around the denominator that defines the updated sample space.
83 verified problems
Waiting-time probabilities, parameter conversions, memorylessness, moments, and insurance-payment applications.
72 verified problems
Count probabilities, rate scaling, aggregation, thinning, and links between Poisson counts and waiting times.
62 verified problems
Standardization, inverse percentiles, linear combinations, normal approximations, and tail control.
57 verified problems
Second moments, transformations, sums, covariance terms, and conditional variance decompositions.
50 verified problems
Continuous and discrete uniform models, interval geometry, transformations, order statistics, and moments.
46 verified problems
Tail-integral proof, survival-function expectation, discrete tail sums, loss payments, linearity, and conditioning.
