## Probability

### The Analysis of Data, volume 1

Basic Definitions: Exercises
$
\def\P{\mathsf{P}}
\def\R{\mathbb{R}}
\def\defeq{\stackrel{\tiny\text{def}}{=}}
\def\c{\,|\,}
$

## 1.9. Exercises

- 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 $\Omega$?
- Can there be a classical probability model on sample spaces that are countably infinite? Provide an example or prove that it is impossible.
- Complete the proof of Proposition 1.5.1.
- 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?
- 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)?
- 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*}
- Prove that $\P_E(A)=\P(A \c E)$ is a probability function if $\P(E)\neq 0$.
- 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.
- Repeat the previous exercise, if the results are observed with order.
- 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?