Associate of the Society of Actuaries (ASA/FSA) Exam P — Probability Syllabus

Exam P — Probability syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Exam P — Probability in Associate of the Society of Actuaries (ASA/FSA), not a summary of it.

1. 1

   Foundations of Probability

   5 topics

   • Set Theory and Sample Spaces
     • Sample spaces, outcomes, and events
     • Unions, intersections, complements, and De Morgan's laws
     • Mutually exclusive and exhaustive events
   • Axioms and Basic Probability Rules
     • Kolmogorov axioms
     • Addition rule and inclusion-exclusion
     • Complement and difference rules
   • Combinatorial Probability
     • Permutations and combinations
     • Multinomial counting
     • Equally likely outcomes and counting arguments
   • Conditional Probability and Independence
     • Definition of conditional probability
     • Multiplication rule and chain rule
     • Independent vs. mutually exclusive events
   • Bayes' Theorem and Law of Total Probability
     • Partitioning the sample space
     • Prior and posterior probabilities
     • Tree-diagram applications
2. 2

   Univariate Random Variables

   4 topics

   • Discrete Random Variables
     • Probability mass functions
     • Cumulative distribution functions
     • Expectation, variance, and moments
   • Continuous Random Variables
     • Probability density functions
     • CDFs and quantiles/percentiles
     • Expectation and variance via integration
   • Moment Generating Functions
     • Deriving moments from the MGF
     • Uniqueness and distribution identification
   • Functions and Transformations of a Random Variable
     • CDF method
     • Change-of-variable (Jacobian) method
     • Expectation of a function of a random variable
3. 3

   Common Distributions

   3 topics

   • Discrete Families
     • Bernoulli and binomial
     • Geometric and negative binomial
     • Poisson and hypergeometric
   • Continuous Families
     • Uniform and exponential
     • Gamma and beta
     • Normal and lognormal
   • Distribution Relationships
     • Poisson limit of the binomial
     • Memorylessness of the exponential
     • Sums of exponentials and the gamma
4. 4

   Multivariate Probability and Limit Theorems

   4 topics

   • Joint, Marginal, and Conditional Distributions
     • Joint pmf/pdf and marginals
     • Conditional distributions and independence
     • Joint moments and covariance
   • Covariance and Correlation
     • Variance of sums and linear combinations
     • Correlation coefficient interpretation
   • Conditional Expectation
     • Law of total expectation (double expectation)
     • Law of total variance
   • Limit Theorems and Approximations
     • Central Limit Theorem and normal approximation
     • Law of large numbers
     • Continuity correction

Exam P — Probability flashcards for Associate of the Society of Actuaries (ASA/FSA)

19 of 56 cards from the Exam P — Probability deck — real questions with worked answers.

1. What is the formula for the probability of the union of two events (inclusion-exclusion for two sets)?

   $P(A \cup B) = P(A) + P(B) - P(A \cap B)$

2. State De Morgan's Laws for two events $A$ and $B$.

   $(A \cup B)^{c} = A^{c} \cap B^{c}$ and $(A \cap B)^{c} = A^{c} \cup B^{c}$

3. What does it mean for events to be mutually exclusive (disjoint), and what is $P(A \cap B)$ in that case?

   Events cannot occur simultaneously: $A \cap B = \varnothing$, so $P(A \cap B) = 0$ and $P(A \cup B) = P(A) + P(B)$.

4. State the three Kolmogorov axioms of probability.

   1) $P(A) \geq 0$ for any event $A$. 2) $P(S) = 1$ for the sample space $S$. 3) For mutually exclusive events, $P\left(\bigcup_{i} A_{i}\right) = \sum_{i} P(A_{i})$.

5. What is the complement rule for probability?

   $P(A^{c}) = 1 - P(A)$

6. State the inclusion-exclusion formula for three events $A$, $B$, $C$.

   $P(A \cup B \cup C) = P(A) + P(B) + P(C) - P(A \cap B) - P(A \cap C) - P(B \cap C) + P(A \cap B \cap C)$

7. How many ordered arrangements (permutations) of $r$ objects from $n$ distinct objects are there?

   $P(n, r) = \dfrac{n!}{(n-r)!}$

8. How many ways can you choose $r$ objects from $n$ distinct objects when order does not matter (combinations)?

   $\binom{n}{r} = \dfrac{n!}{r!\,(n-r)!}$

9. What is the multinomial coefficient for partitioning $n$ objects into groups of sizes $n_{1}, n_{2}, \dots, n_{k}$?

   $\dfrac{n!}{n_{1}!\,n_{2}!\cdots n_{k}!}$, where $\sum_{i=1}^{k} n_{i} = n$.

10. Define the conditional probability $P(A \mid B)$.

    $P(A \mid B) = \dfrac{P(A \cap B)}{P(B)}$, provided $P(B) > 0$.

11. What is the general multiplication rule for $P(A \cap B)$?

    $P(A \cap B) = P(A \mid B)\,P(B) = P(B \mid A)\,P(A)$

12. What condition defines independence of two events $A$ and $B$?

    $P(A \cap B) = P(A)\,P(B)$, equivalently $P(A \mid B) = P(A)$.

13. State the Law of Total Probability for a partition $B_{1}, \dots, B_{n}$ of the sample space.

    $P(A) = \sum_{i=1}^{n} P(A \mid B_{i})\,P(B_{i})$

14. State Bayes' Theorem for a partition $B_{1}, \dots, B_{n}$.

    $P(B_{k} \mid A) = \dfrac{P(A \mid B_{k})\,P(B_{k})}{\sum_{i=1}^{n} P(A \mid B_{i})\,P(B_{i})}$

15. For a discrete random variable, what two conditions must the probability mass function $p(x)$ satisfy?

    $p(x) \geq 0$ for all $x$, and $\sum_{x} p(x) = 1$.

16. Define the expected value of a discrete random variable $X$.

    $E[X] = \sum_{x} x\,p(x)$

17. State the law of the unconscious statistician (LOTUS) for $E[g(X)]$ in the discrete case.

    $E[g(X)] = \sum_{x} g(x)\,p(x)$

18. Give the two equivalent formulas for the variance of $X$.

    $\operatorname{Var}(X) = E\left[(X - \mu)^{2}\right] = E[X^{2}] - (E[X])^{2}$

19. How do the mean and variance transform under a linear change $Y = aX + b$?

    $E[aX + b] = aE[X] + b$ and $\operatorname{Var}(aX + b) = a^{2}\operatorname{Var}(X)$.

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