Events and logical statements

Note that while an experiment can only produce one outcome (out of many possibilities), this is not so for events. Given two events EE and FF, it is possible that neither of them occur, only one of them, or both.

Example 1

You roll a die. Consider the events

E={1,2}E=\{1,2\}

and

F={2,3}F=\{2,3\}

If the outcome is 11, event EE occurs. If the outcome is 22, both events EE and FF occur, if a 33 occurs, event FF occurs and for any other outcome neither EE nor FF occurs.

Let's say we have the two events EE and FF. Often we are not interested in the occurrence of EE and FF, but rather if they both occur in the same experiment, of if only one of them occurred and not the other, and so on.

Set theory allows us to formulate such statements in a simple way. Indeed, if you revisit the example above, note that EE and FF occurs in the same experiment if the experiment has produced an outcome that is in both events EE and FF. In other words, EE and FF both occur if the outcome is in the intersection between EE and FF.

EE or FF occurs if the outcome is in EE, or in FF, or in both, that is, the outcome is in the union of EE and FF.

And EE does not occur if the outcome is not in EE, that is, the outcome is in the complement E′E^\prime.

Let us summarise:

Summary 1

Consider two events EE and FF of a random experiment. Let us perform the experiment.

  • EE and FF occur if the outcome is in EE and FF, that is, the event E∩FE\cap F occurs.
  • EE or FF occurs if the outcome is in EE or FF, tat is, the event E∪FE\cup F occurs.
  • EE does not occur if the outcome is not in EE, thus in E′E^\prime, that is, the event E′E^\prime occurs.

Note that we do not necessarily need to know the possible outcomes of a random experiment - often we are just dealing with events. See the exercises below.

Exercise 1

We regard the weather forecast for tomorrow as a random experiment. Consider the two events

  • EE="the sun shines"
  • FF="it is windy"

Express the following events using EE and FF, and the set operators intersection, union, and complement.

  1. It is windy and the sun shines.
  2. It is windy or the sun shines.
  3. It is not windy
  4. It is not windy and the sun shines.
  5. It is either windy or the sun shines (but not both).
  6. It is neither windy nor does the sun shine.

Also ... why can EE and FF not be outcomes?

Solution
  1. E∩FE\cap F
  2. E∪FE \cup F
  3. F′F^\prime
  4. F′∩EF^\prime \cap E
  5. (F∪E)∩(F∩E)′(F \cup E) \cap (F \cap E)^\prime or we could also write (E∩F′)∪(E′∩F)(E\cap F^\prime) \cup (E^\prime \cap F)
  6. (E∪F)′(E \cup F)^\prime

EE and FF are no outcomes because both EE and FF can occur in an experiment. But for a random experiment, only one outcome can to occur.

Exercise 2

We regard the prediction if the two student Albert and Beth are late at school as a random experiment. We define the two events

  • EE="Albert is late" and
  • FF="Beth is late".

Express the following events as logical statements:

  1. E′E^\prime

  2. E∩FE \cap F

  3. E∪FE\cup F

  4. E∩F′E\cap F^\prime

  5. (E∩F′)∪(E′∩F)(E\cap F^\prime) \cup (E^\prime\cap F)

  6. (E∩F)′(E\cap F)^\prime

Solution
  1. Albert is in time (is not too late).
  2. Albert and Beth are late.
  3. Albert or Beth are late (one of them or both)
  4. Albert is late and Beth is not late
  5. Either Albert or Beth are late, but not both.
  6. Albert and Beth are not late together.

Mutually exclusive events

Two events EE and FF of a random experiment are called mutually exclusive, if the sets EE and FF are disjoint, that is, they do not have a common outcome. Clearly, at most one of those two events can occur if an experiment is performed.

We can generalise this concept to more than two sets. Three events E,FE, F and GG are called pairwise mutually exclusive, if EE, FF, and GG are pairwise disjoint. As above, none of these events have an outcome in common, and thus at most one of them can occur if the experiment is performed.

It is straightforward to generalise this definition to an arbitrary number of events: events E1,...,EmE_1, ..., E_m are pairwise mutually exclusive, if the sets E1,...,EmE_1,...,E_m are mutually disjoint. As above, at most one of them can occur if the experiment is performed.

Exercise 3

Consider a random experiment with sample space SS, and assume that the subsets E1,...,EmE_1,...,E_m form a partition of SS.

  1. Are the events E1,...,EmE_1,...,E_m pairwise mutually exclusive?
  2. The experiment is performed. Which statement is correct:

    • exactly one of the events E1,...,EmE_1,...,E_m must occurs.

    • more than one of the events E1,...,EmE_1,...,E_m can occur.

    • it is possible that none of the events E1,...,EmE_1,...,E_m occurs.

Solution
  1. Yes
  2. the first statement is correct, the others are wrong.