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

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

Question 3

Write a summary, of around forty words, explaining your current best understanding of what a partition is.

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 this idea.