2.1 Outcomes
Probability models can be applied to any situation in which there are multiple potential outcomes and there is uncertainty about which outcome will occur. Due to the wide variety of types of random phenomena, an outcome can be virtually anything:
- the result of a coin flip
- the results of a sequence of coin flips
- a shuffle of a deck of cards
- the weather conditions tomorrow in your city
- the path of a particular Atlantic hurricane
- the daily closing price of a certain stock over the next 30 days
- a noisy electric signal
- the result of a diagnostic medical test
- a sample of car insurance polices
- the customers arriving at a store
- the result of an election
- the next World Series champion
- a play in a basketball game
And on and on. In particular, an outcome does not have to be a number.
Before the random phenomenon occurs it is unknown which outcome will be the result. When the phenomenon takes place, a particular outcome is observed. The first step in defining a probability model for a random phenomenon is to identify the possible outcomes. The sample space is the collection of all possible outcomes of a random phenomenon.
In simple examples we can describe sample space by listing all possible outcomes. However, constructing a list of all possible outcomes is rarely done in practice. We do so here only to provide some concrete examples of sample spaces. While a random phenomenon always has a corresponding sample space, in most situations the sample space of outcomes is at best only vaguely specified and can not be feasibly enumerated.
Example 2.4 Roll a four-sided die twice, and record the result of each roll in sequence. For example, a 3 on the first roll and a 1 on the second is not the same outcome as a 1 on the first roll and a 3 on the second.
Identify the sample space.
We might be interested in the sum of the two rolls. Explain why it is still advantageous to define the sample space as in the previous part, rather than as just \(2, \ldots, 8\).
Solution. to Example 2.4
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We simply enumerate all the possible outcomes: first roll is a 1 and second roll is a 1, first roll is a 1 and second roll is a 2, etc. The sample space consists of 16 possible ordered pairs of rolls, which we can display in a list or table; see Table 2.1
Yes, we might be interested in the sum of the two dice. But we might also be interested in other things, like the larger of the two rolls, or if at least one 3 was rolled, or the result of the first roll. Knowing just the sum of the rolls does not provide as much information about the outcome of the random phenomenon as the sequence of individual rolls does.
| First roll | Second roll |
|---|---|
| 1 | 1 |
| 1 | 2 |
| 1 | 3 |
| 1 | 4 |
| 2 | 1 |
| 2 | 2 |
| 2 | 3 |
| 2 | 4 |
| 3 | 1 |
| 3 | 2 |
| 3 | 3 |
| 3 | 4 |
| 4 | 1 |
| 4 | 2 |
| 4 | 3 |
| 4 | 4 |
In the previous example, there was a single sample space whose outcomes represented the result of the pair of rolls. In particular, there was not a separate sample space for each of the individual rolls.
Here’s another concrete examples where we can list all the outcomes in the sample space. However, keep in mind that enumerating the sample space is rarely done in practice.
Example 2.5 Consider the matching problem (Example 2.4) with \(n=4\). Label the objects 1, 2, 3, 4, and the spots 1, 2, 3, 4, with spot 1 the correct spot for object 1, etc. Identify an appropriate sample space.
Solution. to Example 2.5
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We can consider each outcome to be a particular placement of objects in the spots. For example, object 3 is placed in spot 1, object 2 in spot 2, object 1 in spot 3, and object 4 in spot 4. So the sample space consists of all the possible arrangments of the object labels into the 4 spots. There are 24 outcomes23; see Table 2.2 Recording outcomes in this way provides more information than if we had chosen the sample space to correspond to, for example, the number of objects that were placed in the correct spot.
| Spot 1 | Spot 2 | Spot 3 | Spot 4 |
|---|---|---|---|
| 1 | 2 | 3 | 4 |
| 1 | 2 | 4 | 3 |
| 1 | 3 | 2 | 4 |
| 1 | 3 | 4 | 2 |
| 1 | 4 | 2 | 3 |
| 1 | 4 | 3 | 2 |
| 2 | 1 | 3 | 4 |
| 2 | 1 | 4 | 3 |
| 2 | 3 | 1 | 4 |
| 2 | 3 | 4 | 1 |
| 2 | 4 | 1 | 3 |
| 2 | 4 | 3 | 1 |
| 3 | 1 | 2 | 4 |
| 3 | 1 | 4 | 2 |
| 3 | 2 | 1 | 4 |
| 3 | 2 | 4 | 1 |
| 3 | 4 | 1 | 2 |
| 3 | 4 | 2 | 1 |
| 4 | 1 | 2 | 3 |
| 4 | 1 | 3 | 2 |
| 4 | 2 | 1 | 3 |
| 4 | 2 | 3 | 1 |
| 4 | 3 | 1 | 2 |
| 4 | 3 | 2 | 1 |
In the previous examples, the sample space was discrete, in the sense that the outcomes could be enumerated in a list (though it could be a very long list). But in many cases, it is not possible to enumerate outcomes in a list, even in principle.
For example, consider the circular spinner (like from a kids game) in Figure 2.1. Imagine a needle anchored at the center of the circle which is spun and eventually lands pointing at a number on the outside of the circle. The values in the picture are rounded to two decimal places, but consider an idealized model where the spinner is infinitely precise and the needle infinitely fine so that any real number between 0 and 1 is a possible outcome. The sample space corresponding to a single spin of this spinner is the interval [0, 1]. There are uncountably many numbers in [0, 1] so it would not be possible to enumerate them in a list. The interval [0, 1] is an example of a continuous sample space.
![A continuous [0, 1] spinner. The values in the picture are rounded to two decimal places, but in the idealized model the spinner is infinitely precise so that any real number between 0 and 1 is a possible outcome.](_graphics/uniform-spinner.png)
Figure 2.1: A continuous [0, 1] spinner. The values in the picture are rounded to two decimal places, but in the idealized model the spinner is infinitely precise so that any real number between 0 and 1 is a possible outcome.
Example 2.6 Consider a version of the meeting problem (Example 2.3) where Regina and Cady will definitely arrive between noon and 1, but their exact arrival times are uncertain. Rather than dealing with clock time, it is helpful to represent noon as time 0 and measure time as minutes after noon, so that arrival times take values in the continuous interval [0, 60].
Describe an appropriate sample space. Hint: it might be easiest to draw a picture.
Solution. to Example 2.6
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We can represent an outcome as a (Regina, Cady) pair of arrival times, each in [0, 60]. For example, the outcome (30, 45) represents Regina arriving at 12:30 and Cady at 12:45, while (45, 0) represents Regina arriving at time (12:45) and Cady at noon. The sample space is the set of all possible pairs. We can visualize the sample space24 as the set of points within the colored square with [0, 60] sides in Figure 2.2.
![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].](probsim-book_files/figure-html/meeting-outcome-plot-1.png)
Figure 2.2: 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].
In the previous example, outcomes were measured on a continuous scale; any real number between 0 and 60 was a possible arrival time. In practice we might round the arrival time to the nearest minute or second, but in principle and with infinite precision any real number in the continuous interval \([0, 60]\) is possible.
Furthermore, even in situations where outcomes are inherently discrete, it is often more convenient to model them as continuous. For example, if an outcome represents the annual salary in dollars of a randomly selected U.S. household, it would be more convenient to model the sample space as the continuous interval25 \([0, \infty)\) rather than discrete intervals like \(\{0, 1, 2, \ldots\}\) or \(\{0, 0.01, 0.02, \ldots\}\). Continuous models are often more tractable mathematically than discrete models.
2.1.1 Summary
The sample space is the set of all possible outcomes of a random phenomenon. Outcomes can take a wide variety of forms. In particular, outcomes do not need to be numbers. Whenever possible, a sample space outcome should be defined to provide the maximum amount of information about the outcome of random phenomenon.
In practice we rarely enumerate the sample space as we did for some of the examples in this section. Nonetheless, there is always some underlying sample space corresponding to all possible outcomes of the random phenomenon. Even though the sample space often is at best vaguely defined (e.g., “tomorrow’s weather conditions”) and plays a background role, it is important to first consider what is possible before determining what is probable. The sample space essentially defines the denominator in probability calculations. Considering the sample space can help distinguish between “what is the probability this happens to me?” and “what is the probability this happens to someone somewhere sometime?” (as discussed in Section 1.5.)
There are 4 objects that could potentially go in spot 1, then 3 objects that could potentially go in spot 2, 2 to spot 3, and 1 left for spot 4. This results in \(4\times3\times2\times1=4! = 24\) possible outcomes. We will see more counting rules later.↩︎
Mathematically we can write the sample space as \([0,60]\times [0,60]=[0,60]^2\), the Cartesian product \(\{(x, y): x \in [0, 60], y \in [0, 60]\}\), the set of ordered pairs whose components take values in \([0, 60]\).↩︎
We could also try \([0, m]\) where \(m\) is some large dollar amount providing an upper bound on the maximum possible salary. But we would need to be sure that \(m\) is large enough so that all possible outcomes are in the sample space \([0, m]\). Without knowing this bound in advance, it is convenient to just choose the unbounded interval \([0, \infty)\). There is really no harm in making the sample space bigger than it needs to be, but you can run into problems if you make it too small.↩︎