1.2 Events
==========

:bolditalic:`Key Terms`

:bolditalic:`Event（事件）`
   A set of outcomes from the sample space.

:bolditalic:`Complement（餘事件）`
   All outcomes in the sample space that are not in an event.

:bolditalic:`Intersection（交集）`
   The outcomes that are in both events.

:bolditalic:`Union（聯集）`
   The outcomes that are in one event, the other event, or both.

:bolditalic:`Mutually exclusive（互斥事件）`
   Two events that cannot happen together.

:bolditalic:`Empty set（空集合）`
   A set with no outcomes. We use the symbol :math:`\varnothing`.


1.2.2 Learning Outcomes
-----------------------

After completing this section, students should be able to:

- explain an event in simple words;
- identify an event as a subset of a sample space;
- use :math:`A^c`, :math:`A \cap B`, and :math:`A \cup B`;
- explain the meanings of ``NOT``, ``AND``, and ``OR``;
- use a Venn diagram to describe events;
- identify mutually exclusive events;
- explain the impossible event and the certain event.

1.2.3 Event
-----------

In the previous section, we asked:

    What can happen?

The answer was the **sample space**.

Now we ask:

    Which outcomes do we care about?

The answer is an **event**.

An event is a set of outcomes from the sample space.

.. note::

   **Event:** a set of outcomes from the sample space.

If the sample space is :math:`S` and an event is :math:`A`, we write

.. math::

   A \subseteq S.

This means that every outcome in :math:`A` must also be in :math:`S`.

1.2.4 Example 1: Roll One Die
-----------------------------

**Experiment:** Roll one six-sided die one time.

The sample space is

.. math::

   S = \{1,2,3,4,5,6\}.

Define event :math:`A` as rolling an even number:

.. math::

   A = \{2,4,6\}.

Define event :math:`B` as rolling a number greater than 3:

.. math::

   B = \{4,5,6\}.

Both events are subsets of the sample space:

.. math::

   A \subseteq S
   \qquad \text{and} \qquad
   B \subseteq S.

.. image:: ../_static/figures/probability/figure_1_2_1_event_as_subset_die.png
   :width: 100%
   :align: center

This figure shows that events :math:`A` and :math:`B` are subsets chosen from the sample space :math:`S`.

We will use these two events to study the main event notation.

1.2.5 Event Notation
--------------------

Three important ideas are ``NOT``, ``AND``, and ``OR``.

.. list-table::
   :header-rows: 1
   :widths: 30 30 40

   * - Meaning
     - Symbol
     - Simple description
   * - NOT :math:`A`
     - :math:`A^c`
     - Outcomes not in :math:`A`
   * - :math:`A` AND :math:`B`
     - :math:`A \cap B`
     - Outcomes in both :math:`A` and :math:`B`
   * - :math:`A` OR :math:`B`
     - :math:`A \cup B`
     - Outcomes in :math:`A`, :math:`B`, or both

1.2.6 Complement: NOT
---------------------

The **complement** of event :math:`A` contains all outcomes in :math:`S`
that are not in :math:`A`.

We write the complement as

.. math::

   A^c.

Using

.. math::

   S = \{1,2,3,4,5,6\}

and

.. math::

   A = \{2,4,6\},

we get

.. math::

   A^c = \{1,3,5\}.

Event :math:`A` means ``even``.

Event :math:`A^c` means ``not even``, so it means ``odd``.

1.2.7 Intersection: AND
-----------------------

The **intersection** of :math:`A` and :math:`B` contains the outcomes that
are in both events.

We write

.. math::

   A \cap B.

Using

.. math::

   A = \{2,4,6\}

and

.. math::

   B = \{4,5,6\},

the common outcomes are 4 and 6.

Therefore,

.. math::

   A \cap B = \{4,6\}.

This means:

    the result is even **and** greater than 3.

1.2.8 Union: OR
---------------

The **union** of :math:`A` and :math:`B` contains the outcomes that are in
:math:`A`, in :math:`B`, or in both events.

We write

.. math::

   A \cup B.

Using the same events,

.. math::

   A \cup B = \{2,4,5,6\}.

This means:

    the result is even **or** greater than 3, or both.

.. note::

   In probability, ``A or B`` includes the case where both :math:`A` and
   :math:`B` happen.

For example, 4 and 6 are in both events. In a set, each outcome is written
only once.

1.2.9 Venn Diagram
------------------

A **Venn diagram** gives a picture of events inside a sample space.

In a Venn diagram:

- the rectangle represents the sample space :math:`S`;
- a circle represents an event;
- the overlap of :math:`A` and :math:`B` represents :math:`A \cap B`;
- the parts inside :math:`A`, :math:`B`, or both represent :math:`A \cup B`;
- the part outside :math:`A` represents :math:`A^c`.

.. image:: ../_static/figures/probability/figure_1_2_2_event_notation_venn.png
   :width: 100%
   :align: center

This figure shows the three main event operations visually: :math:`A^c`, :math:`A \cap B`, and :math:`A \cup B`.

For the die example:

.. list-table::
   :header-rows: 1
   :widths: 28 28 44

   * - Region
     - Outcomes
     - Meaning
   * - :math:`A` only
     - :math:`\{2\}`
     - Even, but not greater than 3
   * - :math:`A \cap B`
     - :math:`\{4,6\}`
     - Even and greater than 3
   * - :math:`B` only
     - :math:`\{5\}`
     - Greater than 3, but not even
   * - Outside :math:`A`
     - :math:`\{1,3,5\}`
     - NOT :math:`A`

1.2.10 Mutually Exclusive Events
--------------------------------

Two events are **mutually exclusive** when they cannot happen together.

This means that they have no common outcomes.

.. note::

   Two events are mutually exclusive if

   .. math::

      A \cap B = \varnothing.

1.2.11 Example 2: Odd vs. Even
------------------------------

For one die roll, define

.. math::

   C = \{1,3,5\},

where :math:`C` is the event of rolling an odd number.

Define

.. math::

   D = \{2,4,6\},

where :math:`D` is the event of rolling an even number.

There is no outcome that is both odd and even.

Therefore,

.. math::

   C \cap D = \varnothing.

So, :math:`C` and :math:`D` are mutually exclusive.

Compare this with the earlier events :math:`A` and :math:`B`:

.. math::

   A \cap B = \{4,6\} \neq \varnothing.

Therefore, :math:`A` and :math:`B` are **not** mutually exclusive.

.. image:: ../_static/figures/probability/figure_1_2_3_mutually_exclusive_events.png
   :width: 100%
   :align: center

This figure shows a mutually exclusive pair of events for one die roll: odd and even.

1.2.12 Special Events
---------------------

Two special events are useful.

The **impossible event** contains no outcomes. It is written as

.. math::

   \varnothing.

For example, rolling 7 on a standard six-sided die is impossible.

The **certain event** is the whole sample space:

.. math::

   S.

For example, when a standard die is rolled, the result must be one of

.. math::

   \{1,2,3,4,5,6\}.

An event and its complement have two important relationships:

.. math::

   A \cap A^c = \varnothing

and

.. math::

   A \cup A^c = S.

The first result means that an event and its complement cannot happen
together.

The second result means that one of them must happen.

1.2.13 A Simple Method
----------------------

When working with events, use these steps:

1. Write the sample space :math:`S`.
2. Define the event or events.
3. Look for the words ``NOT``, ``AND``, or ``OR``.
4. Choose the correct symbol.
5. List the outcomes that match the event.
6. Check whether the events have common outcomes when studying mutually exclusive events.

Use this guide:

.. list-table::
   :header-rows: 1
   :widths: 30 30 40

   * - Word
     - Symbol
     - Meaning
   * - NOT
     - :math:`A^c`
     - Outcomes not in :math:`A`
   * - AND
     - :math:`A \cap B`
     - Outcomes in both events
   * - OR
     - :math:`A \cup B`
     - Outcomes in either event or both

1.2.14 Common Mistakes
----------------------

A common mistake is to include an outcome that is not in the sample space.

For a standard die,

.. math::

   7 \notin S.

Another mistake is to confuse intersection and union.

- Intersection means ``AND``.
- Union means ``OR``.

A third mistake is to think that ``OR`` means only one event can happen.

In probability, ``A or B`` also includes the case where both events happen.

A final mistake is to call two events mutually exclusive when they have a
common outcome.

If

.. math::

   A \cap B \neq \varnothing,

then :math:`A` and :math:`B` are not mutually exclusive.

1.2.15 Summary
--------------

An **event** is a set of outcomes from the sample space.

.. math::

   A \subseteq S.

The complement means ``NOT``.

.. math::

   A^c.

The intersection means ``AND``.

.. math::

   A \cap B.

The union means ``OR``.

.. math::

   A \cup B.

Mutually exclusive events cannot happen together.

.. math::

   A \cap B = \varnothing.

The impossible event is :math:`\varnothing`.

The certain event is :math:`S`.

An event and its complement satisfy

.. math::

   A \cap A^c = \varnothing

and

.. math::

   A \cup A^c = S.

The main idea is:

    Sample Space -> Choose Events -> Use NOT, AND, or OR

.. 1.2.16 Short Review Questions
.. -----------------------------

.. 1. What is an event?
.. 2. What does :math:`A \subseteq S` mean?
.. 3. What does :math:`A^c` mean?
.. 4. What does :math:`A \cap B` mean?
.. 5. What does :math:`A \cup B` mean?
.. 6. For one die, let :math:`A=\{2,4,6\}`. Find :math:`A^c`.
.. 7. Let :math:`A=\{2,4,6\}` and :math:`B=\{4,5,6\}`. Find :math:`A \cap B`.
.. 8. Let :math:`A=\{2,4,6\}` and :math:`B=\{4,5,6\}`. Find :math:`A \cup B`.
.. 9. What does it mean when two events are mutually exclusive?
.. 10. Give one example of two mutually exclusive events for a die roll.
.. 11. What is the impossible event?
.. 12. What is the certain event?