Events as Subsets of the Sample Space

From Possible Outcomes to Events

In Sample Spaces and Outcomes, we described the results of rolling two dice as ordered pairs: the first number records the first die, and the second number records the second die. That sample space contains 36 outcomes. A question about a chance process usually focuses not on every possible result, but on results that meet a particular condition. For example, we might ask whether the sum of the two dice is 8.

The condition “the sum is 8” selects some outcomes from the full sample space. The selected outcomes form an event. Thinking of an event as a set makes it clear what counts as happening: the event occurs when the actual outcome belongs to that set.

We can use \(S\) for the sample space and a capital letter such as \(A\) or \(E\) for an event. The notation \(E \subseteq S\) says that every outcome in event \(E\) is also an outcome in sample space \(S\). An event does not add new possible results; it selects some of the results already in the sample space.

The word “favorable” has a specific, neutral meaning when counting outcomes: a favorable outcome is one that meets the event’s condition. It does not mean that the result is personally desirable. If the event is “the sum is 8,” then an ordered pair is favorable exactly when its two entries add to 8.

Translate a Condition into a Set

To describe an event, start with the sample space and apply the condition to each possible outcome. Keep an outcome if it satisfies the condition; leave it out if it does not. The result is the event set. For a small sample space, listing the set is often the clearest approach.

A verbal condition can also be written using set-builder notation. For example, \(\{(d_1,d_2)\in S: d_1+d_2=8\}\) means “the set of ordered pairs in \(S\) for which the first number plus the second number equals 8.” The colon means “such that.” Set-builder notation states a rule for membership; listing the outcomes makes the members visible.

The event’s size is the number of outcomes it contains. In this tutorial, counting the outcomes that satisfy a condition is the goal; that count is not itself a probability. To use a count to describe chance, we will also need to consider how likely the sample-space outcomes are. The count is an important first step, but it answers a different question from “How likely is the event?”

Count Outcomes by Applying the Rule

Some event conditions use phrases such as “at least,” “more than,” or “exactly.” Translate the phrase carefully before listing outcomes. “At least 10,” for example, includes 10 and every larger value in the stated range. It does not mean only values greater than 10.

An organized list reduces the chance of missing an outcome. For two dice, group ordered pairs by their first-die result, just as the sample space was organized in the earlier tutorial. For each first-die result, identify which second-die results meet the condition. Then count the entries in the event set.

Events in Other Sample Spaces

The idea of an event applies to any sample space, not only dice. In Sample Spaces and Outcomes, three coin flips were represented by sequences of \(H\) and \(T\). A condition on the number or order of heads selects a subset of those sequences. As with dice, first be precise about what one outcome records.

An event may contain just one outcome, several outcomes, all the outcomes, or none of them. If a condition identifies one particular sequence, its event contains one outcome. If every outcome meets the condition, the event is the entire sample space. If no possible outcome meets it, the event is the empty set, written \(\varnothing\). These are all valid sets of outcomes.

When an Event Has One or No Outcomes

A useful way to check your understanding is to consider conditions at the edges of what the process can produce. For two six-sided dice, the event “the sum is 12” contains only \((6,6)\). The event “the sum is 1” contains no outcomes, because the smallest possible sum is 2. The event “the sum is between 2 and 12, inclusive” contains every outcome in \(S\).

These examples show why an event must be checked against the sample space. A condition might sound meaningful but still be impossible under the stated process. Or it might be broad enough to include every possible outcome. In either case, the event is still a subset of \(S\): the empty set is a subset, and the whole sample space is a subset of itself.

Common Mistakes and AP Exam Tips

• Giving the event as a value instead of a set of outcomes. For two dice, “8” is the sum value, not the full event in the ordered-pair sample space. State the pairs that produce that sum.
• Leaving out reversed pairs. If the outcome records first die and second die, \((2,6)\) and \((6,2)\) are distinct. Include both whenever both satisfy the condition.
• Counting a repeated entry twice. The pair \((4,4)\) is one outcome. It does not become two outcomes because the same number appears on both dice.
• Misreading “at least” or “exactly.” “At least 10” includes a sum of 10; “exactly two heads” excludes sequences with one or three heads. Rewrite the condition in a precise way before listing.
• Including outcomes that fail the condition. Check each listed outcome against the event rule. For a sum event, add the coordinates; for a coin-flip event, count the required symbols.
• Calling the number of favorable outcomes the probability. A count describes how many outcomes are in the event. Probability also depends on the chance model and how likely the outcomes are. Keep those answers separate.

For a full-credit response, identify how outcomes are recorded, state the event condition, and give the set or a clear count of all outcomes that satisfy it. When the outcomes are ordered, preserve their order in the list. A short verification—such as checking each sum or counting each sequence’s heads—helps show that the event is complete and contains no extra outcomes.