Raising the bar

Chapter 4. Probability

Unit 4.10. Questions for repetition

1. What do we mean by set operations?
2. Prove de Morgan laws.
3. Prove and illustrate on a Venn diagram equation Eq. (4.2) (see Probability).
4. Prove $latex P(A \cup B) = P(A) + P(B) - P(A \cap B)$.
5. Prove the formulas for orderings and combinations.
6. Using a simple example, in one block give all definitions and identities related to joint and marginal probabilities.
7. Solve Exercise 4.7 (see Conditional probabilities).
8. Derive the complex form of the Bayes theorem.
9. What can you say about event $latex A$ if $latex P(A) = {P^2}(A)$?
10. If $latex A$ and $latex B$ are independent and $latex B$ and $latex C$ are independent, then $latex A$ and $latex C$ are also independent (transitivity). True or false?
11. There are only two events that are independent of themselves: the empty event and the sample space. True or false?
12. If $latex A$ and $latex B$ are independent and both have positive probability, then they cannot be mutually exclusive. True or false?
13. Any event is independent of its complement. True or false?
14. If $latex A,B$ cover the whole sample space $latex S$, $latex A \cup B = S$, then $latex P(A|C) + P(B|C) \ge 1$.
15. If $latex A$ is independent of $latex B$ and $latex C$, then conditioning $latex A$ on either of $latex B,C$ yields the same result.

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