2.2 Events
An event is something that might happen. For example, if we’re interested in the weather conditions in our city tomorrow, events include
- the high temperature is 75°F
- the high temperature is above 75°F
- it rains
- it does not rain
- it rains and the high temperature is above 75°F
An outcome consists of all the information about tomorrow’s weather conditions, while an event is a collection of outcomes that satisfy some criteria.
The sample space is the collection of all possible outcomes; an event represents only those outcomes which satisfy some criteria.
Events are typically denoted with capital letters near the start of the alphabet, with or without subscripts (e.g. \(A\), \(B\), \(C\), \(A_1\), \(A_2\)).
Mathematically, events are sets, so events can be composed from others using basic set operations like unions (\(A\cup B\)), intersections (\(A \cap B\)), and complements (\(A^c\)).
- Complements. Read \(A^c\) as “not \(A\)”, the outcomes that do not satisfy \(A\)
- Intersections. Read \(A\cap B\) as “\(A\) and \(B\)”, the outcomes that satisfy both \(A\) and \(B\)
- Unions. Read \(A \cup B\) as “\(A\) or \(B\)”, the outcomes that satify \(A\) or \(B\). Note that unions (\(\cup\), “or”) are always inclusive: \(A\cup B\) occurs if \(A\) occurs but \(B\) does not, \(B\) occurs but \(A\) does not, or both \(A\) and \(B\) occur.
If the outcomes of a sample space are represented by rows in a table, then events are collections (or subsets) or rows which satisfy some criteria.
Example 2.7 Roll a four-sided die twice, and record the result of each roll in sequence. Using the sample space from Example 2.4, identify the following events.
- \(A\), the event that the sum of the two dice is 4.
- \(B\), the event that the sum of the two dice is at most 3.
- \(C\), the event that the larger of the two rolls (or the common roll if a tie) is 3.
- \(A\cap C\) (identify and interpret).
- \(D\), the event that the first roll is a 3.
- \(E\), the event that the second roll is a 3.
- \(D \cap E\) (identify and interpret).
- \(D \cup E\) (identify and interpret).
- If the outcome is \((1, 3)\), which of the events above occurred?
Solution. to Example 2.7
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Remember that the sample space consists of 16 possible ordered pairs of rolls, (first roll, second roll); see 2.1. All events must be defined as subsets of this sample space.
- \(A\) consists of the outcomes (1, 3), (2, 2), and (3, 1). Mathematically, event \(A\) is the set \(\{(1, 3), (2, 2), (3, 1)\}\). This event is highlighted in 2.3.
- \(B\) consists of the outcomes (1, 1), (1, 2), and (2, 1).
- \(C\) consists of the outcomes (1, 3), (2, 3), (3, 1), (3, 2), and (3, 3).
- \(A\cap C\), which consists of the outcomes (1, 3) and (3, 1), is the event that both the sum of the two dice is 4 and the larger of the two rolls is 3.
- Each outcome in the sample space consists of a pair of rolls, so we must account for both rolls in defining events, even if the event of interest involves just the first roll. (Remember, there is always a single sample space upon which all events are defined.) So \(D\) consists of the outcomes (3, 1), (3, 2), (3, 3), and (3, 4).
- \(E\) consists of the outcomes (1, 3), (2, 3), (3, 3), and (4, 3). Note that this is not the same event as \(D\).
- \(D \cap E\), which consists only of the outcome (3, 3), is the event that both rolls result in a 3. While an event is always a set, it can be a set consisting of a single outcome (or the empty set).
- \(D \cup E\), which consists of the outcomes (3, 1), (3, 2), (3, 3), (3, 4), (1, 3), (2, 3), and (4, 3) is the event that at least one of the two rolls results in a 3. Notice that the union is inclusive: \((3, 3)\), the outcome that satisfies both \(D\) and \(E\), is an element of \(D\cup E\). But also notice that the outcome \((3, 3)\) only appears once.
- If the outcome is \((1, 3)\) then events \(A\), \(C\), \(A\cap C\), \(E\), \(D\cup E\) all occur. Events \(B,\) \(D\) and \(D\cap E\) do not occur.
| First roll | Second roll | Sum is 4? |
|---|---|---|
| 1 | 1 | no |
| 1 | 2 | no |
| 1 | 3 | yes |
| 1 | 4 | no |
| 2 | 1 | no |
| 2 | 2 | yes |
| 2 | 3 | no |
| 2 | 4 | no |
| 3 | 1 | yes |
| 3 | 2 | no |
| 3 | 3 | no |
| 3 | 4 | no |
| 4 | 1 | no |
| 4 | 2 | no |
| 4 | 3 | no |
| 4 | 4 | no |
We reiterate (again!) that there is a single sample space, upon which all events are defined. In the above example, events that involved only the first or second roll such as \(D\) and \(E\) were still defined in terms of pairs of rolls. An outcome in a sample space should be defined to record as much information as possible so that the occurrence or non-occurrence of all events of interest can be determined.
Some events consist of a single outcome, or no outcomes at all (\(\emptyset\)). Events \(A_1, A_2. A_3, \ldots\) are disjoint (a.k.a. mutually exclusive) if they have no outcomes in common; that is, if \(A_i \cap A_j = \emptyset\) for all \(i\neq j\). Roughly, disjoint events do not “overlap”. In the previous example, events \(B\) and \(C\) are disjoint; there are no outcomes for which both the sum of the dice is at most 3 and the larger roll is a 3. (Mathematically, \(B\cap C \ emptyset\), the empty set.)
Example 2.8 objects labeled 1, 2, 3, 4, are placed at random in spots labeled 1, 2, 3, 4, with spot 1 the correct spot for object 1, etc. Using the sample space from Example 2.5, identify the following events.
- \(A\), the event that all objects are put in the correct spot.
- \(B\), the event that no objects are put in the correct spot.
- \(C\), the event that exactly 3 objects are put in the correct spot.
- \(A_3\), the event that object 3 is put (correctly) in spot 3.
Solution. to Example 2.8
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Recall that each outcome is a particular placement of objects in the spots. For example, the outcome (3, 2, 1, 4) — which we’ll shorten to 3214 — signifies that object 3 is put in spot 1, object 2 in spot 2, object 1 in spot 3, and object 4 in spot 4.
- There is only one outcome, 1234, for which all objects are put in the correct spot, so \(A=\{1234\}\). Remember that an event is always a set, but it can be a set consisting of a single outcome.
- For each outcome in the sample space check to see if the criteria holds to identify \(B=\{2143, 2341, 2413, 3142, 3412, 3421, 4123, 4312, 4321\}\) as the event that no objects are put in the correct spot.
- There are no outcomes in which exactly 3 objects are put in the correct spot so \(C=\emptyset\). (If three objects are in their correct spots, then the remaining object must be in its correct spot too.)
- \(A_3=\{1234, 1432, 2134, 2431, 4132, 4231\}\) is the event that object 3 is put (correctly) in spot 3. This event is highlighted in 2.4. Even though event \(A_3\) only concerns object 3, since the sample space consists of the placements of each of the objects then all events must be expressed in terms of these outcomes.
| Spot 1 | Spot 2 | Spot 3 | Spot 4 | Object 3 in spot 3? |
|---|---|---|---|---|
| 1 | 2 | 3 | 4 | yes |
| 1 | 2 | 4 | 3 | no |
| 1 | 3 | 2 | 4 | no |
| 1 | 3 | 4 | 2 | no |
| 1 | 4 | 2 | 3 | no |
| 1 | 4 | 3 | 2 | yes |
| 2 | 1 | 3 | 4 | yes |
| 2 | 1 | 4 | 3 | no |
| 2 | 3 | 1 | 4 | no |
| 2 | 3 | 4 | 1 | no |
| 2 | 4 | 1 | 3 | no |
| 2 | 4 | 3 | 1 | yes |
| 3 | 1 | 2 | 4 | no |
| 3 | 1 | 4 | 2 | no |
| 3 | 2 | 1 | 4 | no |
| 3 | 2 | 4 | 1 | no |
| 3 | 4 | 1 | 2 | no |
| 3 | 4 | 2 | 1 | no |
| 4 | 1 | 2 | 3 | no |
| 4 | 1 | 3 | 2 | yes |
| 4 | 2 | 1 | 3 | no |
| 4 | 2 | 3 | 1 | yes |
| 4 | 3 | 1 | 2 | no |
| 4 | 3 | 2 | 1 | no |
When more than just a few events are of interest, subscripts are commonly used to identify different events. In the previous example, we might also be interested in \(A_1\), the event that object 1 is placed in spot 1; $A_2, the event that object 2 is placed in spot 2; and so on.
Remember that intervals of real numbers such as \((a,b), [a,b], (a,b]\) are also sets, and so can also be events. For example, if an outcome is the result of a single spin of the spinner in 2.1, events include
- \([0, 0.5]\), the result is between 0 and 0.5 (the needle lands in the right half of the spinner)
- \([0.75, 1]\), the result is between 0.75 and 1 (the needle lands in the northwest quarter of the spinner)
- \([0.595, 0.605)\), the result rounded to two decimal places is 0.6
- \(\{0.6\}\), the result is 0.6 exactly (the needle points exactly at 0.60000000\(\ldots\))
It is often helpful to conceptualize and visualize events (sets) with pictures, especially when dealing with continuous sample spaces.
Example 2.9 Using the sample space from Example 2.6, identify the following events using pictures. (Hint: start with Figure 2.2.)
- Identify \(A\), the event that Regina arrives after Cady.
- Identify \(B\), the event that either Regina or Cady arrives before 12:30.
- Identify \(C\), the event that Cady arrives first and Regina arrives at most 15 minutes after Cady.
- Identify \(D\), the event that Regina arrives before 12:24.
Solution. to Example 2.9
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![The square represents the sample space in Example 2.6. Each point within the square is a (Regina, Cady) pair of arrival times in [0, 60]. Events from 2.9 are shaded in orange.](probsim-book_files/figure-html/fig-uniform-event-1.png)
In the previous example, the sample space consists of (Regina, Cady) pairs of arrival times so any event must be expressed as a collection of pairs. Even though the criteria for event \(D\) involves only Regina’s arrival time, the event is not simply [0, 24]; we need to consider all (Regina, Cady) pairs for which the Regina component is in the interval [0, 24].
In many situations it is not possible to explicitly define a sample space in a compact way, and so outcomes and events are often only vaguely defined. Nevertheless, there is always a sample space in the background representing possible outcomes, and collections of these outcomes represent events of interest.
2.2.1 Summary
- The sample space is the set of all possible outcomes.
- An event is a collection of outcomes that satisfy some particular criteria. That is, an event is a subset of the sample space,
- All events of interest are defined in terms of a single sample space.
- An event can be a set consisting of a single outcome, or no outcomes at all (the empty set).
- Events can be composed from others using basic set operations like intersections (\(A \cap B\), “\(A\) and \(B\)”), unions (\(A \cup B\), “\(A\) or \(B\)”), and complements (\(A^c\), “not \(A\)”)
- Tables, lists, and pictures can be used to conceptualize and visualize events.