Maths for Economics 6: Probability and Economics Topic 1 - Uncertainty

Question 1

Describe the relationship between the ‘Theory of Probability’ and ‘Theory of Uncertainty’

Answer 1

The Theory of Probability:
 Is a special case of the Theory of Uncertainty
 It makes relatively demanding assumptions. (The existence
of probability numbers.)
 applies to a relatively narrow (but still huge!) range of cases
 but is far richer and more powerful in conclusions than the
general theory of uncertainty.

Question 2

I am uncertain on which day of the week I will die. What is the
sample space for this experiment?

Answer 2

Ω = {𝑆𝑢𝑛𝑑𝑎𝑦, 𝑀𝑜𝑛𝑑𝑎𝑦, 𝑇𝑢𝑒𝑠𝑑𝑎𝑦, 𝑊𝑒𝑑𝑛𝑒𝑠𝑑𝑎𝑦,
𝑇ℎ𝑢𝑟𝑠𝑑𝑎𝑦, 𝐹𝑟𝑖𝑑𝑎𝑦, 𝑆𝑎𝑡𝑢𝑟𝑑𝑎𝑦}.

Question 3

How do we go about answering an ‘uncertainty’ question?

Include relevant definitions

Answer 3

1. Make a complete list of all possible outcomes, assigning each outcome to Greek letter. In making such lists, choose outcomes so that they are:
    exhaustive (one of them must happen);
    mutually exclusive (not more than one can happen).
   In other words, compile the list so that only & exactly one outcome must
   occur.
   The possibilities are called ‘simple outcomes’ or
   ‘elementary outcomes’. Such a list is called a sample space. It is a set, usually denoted by capital Greek omega
   ‘Space’ is a fancy mathematical word, used to discourage outsiders.
   Think of a ‘sample space’ as a simple outcome list, or possibility set.
2. We can use curly brackets (braces) borrowed from set theory, and
   write omega = {list of all outcomes}. The sample space is the set of simple
   outcomes

Question 4

What’s an ‘event’?

Answer 4

An event is defined as: a set of simple outcomes. So we can also define
it as a subset of the sample space (which is itself an event).
Just as every event yields a set of simple outcomes, so every set of
simple outcomes defines some event. All simple outcomes are events, but not all events are simple
outcomes

Question 5

I am uncertain on which day of the week I will die. For this
experiment, give an example of an event that is not a simple
outcome.

Answer 5

For instance, the event 𝐸 = {𝑇𝑢𝑒𝑠𝑑𝑎𝑦, 𝑊𝑒𝑑𝑛𝑒𝑠𝑑𝑎𝑦} that I will die on
a day that contains the letter ‘e’.

Question 6

Define ‘simple outcome’ in terms of ‘event’.

Answer 6

A simple outcome is an event containing
just one element of the sample space