4  Questions for Chapter 4

4.1 Stretch

Exercise 4.1 Get some practice in thinking about subjective probability by specifying your own subjective probabilities for your final grade in 1H, and another event of concern to you (your choice, but try to make it interesting!). Explain how you decided your probability values.

Remember, these are your own subjective values, so as long as they obey the probability axioms, they cannot be wrong; however, your reasoning can, of course, be faulty. For instance, you cannot get first class marks in all courses and then have a second class overall mark.

Discuss your solution with your tutor.

Exercise 4.2 Provide an intuitive justification for A3, namely \[ \pr{A \cup B} = \pr{A} + \pr{B}\text{ whenever }A\cap B=\emptyset \] under

1. the relative frequency interpretation, and
2. the betting interpretation.

In other words, write down the definitions of the three probabilities \(\mathbb{P}(A)\), \(\mathbb{P}(B)\), and \(\mathbb{P}(A \cup B)\) under these interpretations, and show, for each interpretation, why A3 should hold. For the betting interpretation, you may assume that a combination of fair bets is also fair.