Charta is on Google Play: practise on the go. Get the app
Charta

State the meaning of an event and of the event space within which events are defined.

FRM® Part I · Quantitative Analysis · Fundamentals of Probability · FRM1.QA.1.a

The idea

A probability model starts with the sample space, the set of every outcome that could occur, for example the possible year-end ratings of a bond. An event is any subset of the sample space: a single outcome or a collection of outcomes, such as the bond is downgraded, which gathers every rating below today's. The event space is the collection of events to which probabilities are assigned. To be coherent it must contain the sample space itself. It must also contain the complement of every event it contains. And it must contain the union of any countable collection of events it contains. These rules guarantee that statements such as not A and A or B always have a probability. If the sample space has n outcomes and every subset counts as an event, there are 2ⁿ events, including the empty set. The usual error is to confuse an outcome with an event, or the sample space with the event space. On the exam, count subsets and test closure.

Formulas

Number of events when every subset is an event

Events = 2ⁿ

n
the number of outcomes in the sample space

Holds when every subset of the sample space counts as an event. The count includes the empty set and the sample space itself.

A practice question

From the question bank, with the reason each wrong option is wrong.

In a probability model for the year-end rating of a bond, which statement correctly describes an event?

  1. Any subset of the sample space, whether a single outcome or a collection of outcomes, to which a probability can be assigned - correct
  2. A single outcome of the experiment; a group of outcomes is called a sample spaceWhy not: Treats an event as one outcome only and mislabels a collection of outcomes as a sample space.
  3. The probability number between 0 and 1 attached to an outcomeWhy not: Confuses the event with the probability assigned to it.
  4. An outcome that is certain to occur and therefore has probability oneWhy not: Describes only the sample space, the certain event; most events are neither certain nor impossible.

An event is a subset of the sample space: one outcome or a collection of outcomes, such as 'downgraded', that groups several ratings. It is a set of outcomes, not a probability and not a single outcome only.

A harder one

A credit model has a sample space of four mutually exclusive year-end states for a counterparty: upgrade, unchanged, downgrade and default. If every subset of the sample space is treated as an event, how many events are in the event space?

  1. 4Why not: Counts the four outcomes instead of the subsets (events) that can be formed from them.
  2. 15Why not: Excludes the empty event, which is always a member of the event space.
  3. 8Why not: Uses 2^(n-1) instead of 2^n, so it counts only half of the subsets.
  4. 16 - correct

Each of the 4 outcomes is either in or out of a subset, giving 2^4 = 16 subsets, including the empty set and the whole sample space. Every one of these is an event.

The hardest tier

A die is rolled, so the sample space is {1, 2, 3, 4, 5, 6}. Writing E for the empty event and S for the whole sample space, which collection of events is a valid event space?

  1. E, {1,2}, SWhy not: {1,2} is included but its complement {3,4,5,6} is missing, so the collection is not closed under complements.
  2. E, {1,2}, {3,4}, {5,6}, SWhy not: Complements such as {3,4,5,6} and unions such as {1,2,3,4} are missing, so it is not closed under complements or unions.
  3. E, {1,2}, {3,4,5,6}, S - correct
  4. {1,2} and {3,4,5,6} onlyWhy not: Omits the sample space and the empty event, although the union of the two sets is S and so must be included.

A valid event space contains S, and the complement and union of its members. E, {1,2}, {3,4,5,6}, S satisfies all of this: the complement of {1,2} is {3,4,5,6}, their union is S, and the complement of S is E.

In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.

Start free →

More in Fundamentals of Probability