## Probability

### The Analysis of Data, volume 1

Basic Definitions: Exercises

## 1.9. Exercises

1. Extend the argument at the end of Section 1.2 and characterize probability functions on a countably infinite $\Omega$ using a sequence of non-negative numbers that sum to one. What is the problem with extending this argument further to uncountably infinite $\Omega$?
2. Can there be a classical probability model on sample spaces that are countably infinite? Provide an example or prove that it is impossible.
3. Complete the proof of Proposition 1.5.1.
4. Describe a sample space consistent with the experiment of drawing a hand in poker. Write the events $E$ corresponding to drawing three aces and drawing a full house (and their sizes $|E|$). What is the event corresponding to the intersection of the two events above, and what is its size and probability under the classical model?
5. Formulate a theory of probability that mirrors the standard theory, with the only difference that the second axiom would be $\P(\Omega)=2.$ How would the propositions throughout the chapter change (if at all)?
6. Show a situation where we have three events that are independent but not mutually independent. Hint: Look for a probability function satisfying \begin{align*} \P(A)&=\P(B)=\P(C)=1/3\\ \P(A\cap B)&=\P(A\cap C)=\P(B\cap C)=1/9=\P(A\cap B\cap C). \end{align*}
7. Prove that $\P_E(A)=\P(A \c E)$ is a probability function if $\P(E)\neq 0$.
8. Consider the experiment of throwing three fair six-sided dice independently and observing the results without order. Identify the sample space, and the most and least probable elements of it.
9. Repeat the previous exercise, if the results are observed with order.
10. Generalize the Principle of Inclusion-Exclusion to a union of three sets. Can you further generalize it to a union of an arbitrary number of sets?