Assignment 1

Question 1 (Exercise 1.14)

A special case of (C8) (Boole’s inequality) is that for any finite collections of sets \(A_1, A_2, \ldots, A_k\), \[ \mathbb{P}( \bigcup_{i=1}^k A_i ) \leq \sum_{i=1}^k \mathbb{P}(A_i) .\] Prove this result directly, using just (A1) and (C6), by induction.

Question 2 (Exercise 1.23)

Suppose \(A\), \(B\), \(C\) are events with \[\begin{align*} \mathbb{P}(A) &= 0.5, & \mathbb{P}(B)&= 0.7, & \mathbb{P}(C) &= 0.6, \\ \mathbb{P}(A\cap B) &= 0.3, & \mathbb{P}(B\cap C)&= 0.4, & \mathbb{P}(C \cap A) &= 0.2, \\ \mathbb{P}( A \cap B \cap C) &= 0.1. \end{align*}\]

Find the probability that exactly two of \(A\), \(B\), \(C\) occur.
Find the probability that exactly one of \(A\), \(B\), \(C\) occurs.

Question 3

Consider a Probability 1 lecture, with one lecturer and up to 320 students in the same room.

Describe a probability scenario and probability space that one might observe in this lecture. You should explain:

the scenario;
its sample space of possible outcomes;
the \(\sigma\)-algebra of events in which you are interested;
and how you would expect the probability measure to work.

Next week, you will be asked to look at pairs of these summaries, submitted by your peers, and select the summary which you think best encompasses the concepts you’ve learned so far. Creativity and good explanations will be rewarded.