Read the Event Before You Read the Probability
Probability notation is a compact way to describe chance questions, but a small change in the symbols can change the event being discussed. In the previous tutorial, Choosing Between Counting and Formula Approaches, you used events and probability rules to solve problems. Here the goal is to read and write the notation accurately before choosing a calculation.
An event is a set of outcomes, often given a capital letter such as \(A\). The expression \(P(A)\) is the probability that event \(A\) occurs. The letter names the event; \(P\) tells us to consider its probability. These are different kinds of objects: \(A\) describes outcomes, while \(P(A)\) is a number from 0 to 1.
The words inside the parentheses identify the event whose probability is being requested. For instance, \(P(A\cap B)\) is not the probability of \(A\) plus the probability of \(B\). It describes the overlap: outcomes in which both events occur. As discussed in Drawing a Venn Diagram of Two Events, that overlap is the intersection \(A\cap B\).
Likewise, “not” changes the event before we take its probability. \(A^c\) is the complement of \(A\), the event that \(A\) does not occur. The complement rule from The Complement Rule gives a convenient way to calculate its probability:
A reliable habit is to say the expression in words before calculating it. Read \(P(A)\) as “probability of A,” \(P(A\cap B)\) as “probability of both A and B,” and \(P(A^c)\) as “probability of not A.” This brief translation helps keep the event, its probability, and the calculation separate.
Worked Example: Identifying an Overlap in a Table
Worked Example: Identifying an Overlap in a Table
A recreation center asks 120 visitors whether they brought a reusable water bottle and whether they bought a snack. Let \(A\) be the event that a randomly selected visitor brought a reusable bottle. Let \(B\) be the event that the visitor bought a snack. The results are shown below.
| Bought a snack | Did not buy a snack | Total | |
|---|---|---|---|
| Brought a bottle | 30 | 42 | 72 |
| Did not bring a bottle | 15 | 33 | 48 |
| Total | 45 | 75 | 120 |
Find \(P(A)\), \(P(B)\), and \(P(A\text{ and }B)\). In each case, a visitor is selected at random from the full group of 120 visitors, so the denominator is 120.
Probability of A: The event \(A\) includes everyone in the “Brought a bottle” row, whether or not the visitor bought a snack. There are 72 such visitors.
Probability of B: The event \(B\) includes everyone in the “Bought a snack” column, whether or not the visitor brought a bottle. There are 45 such visitors.
Probability of A and B: Both events occur for a visitor in the table cell where the “Brought a bottle” row and “Bought a snack” column meet. That cell contains 30 visitors.
Conclude: For a randomly selected visitor, the probability of bringing a bottle is 0.600, the probability of buying a snack is 0.375, and the probability of doing both is 0.250. The phrase “and” directs us to the shared cell, not to either full margin.
This example also shows why \(P(A)\) and \(P(A\cap B)\) do not mean the same thing. \(P(A)\) counts all visitors who brought a bottle, including the 42 who did not buy a snack. \(P(A\cap B)\) counts only the visitors who brought a bottle and bought a snack. When a two-way table is available, check whether the notation refers to a row or column total or to an interior cell, as in Using Two-Way Tables to Find Probabilities.
“Not A” Means the Complement of the Event
The phrase “not \(A\)” refers to all outcomes in the sample space that are outside event \(A\). It does not mean that the probability itself is somehow made negative. If \(A\) is the event “a package arrives late,” then “not \(A\)” means “the package does not arrive late.” It is still an event, and \(P(\text{not }A)\) is its probability.
When the probability of \(A\) is known, the complement rule can find the probability of not \(A\). When counts are available, you can also count the outcomes outside \(A\) directly. These are two ways to calculate the same probability, provided they use the same sample space and event definition.
Worked Example: A Package That Is Not Late
In a modeled shipment group of 200 packages, 18 arrive late. Select one package at random. Let \(L\) be the event that the package arrives late. Find the probability that the package is not late.
Using the complement rule: First calculate the probability of \(L\), then subtract it from 1.
Check by counting: Of the 200 packages, \(200-18=182\) are not late. The direct count gives the same result.
Conclude: Under this model, the probability that a randomly selected package is not late is 0.910. In the notation, \(P(\text{not }L)\) and \(P(L^c)\) name the same probability.
A common notation error is to write “not \(P(L)\)” as if it were the complement event. The complement applies to the event \(L\), so the standard notation is \(P(L^c)\), or \(P(\text{not }L)\). The number \(1-P(L)\) is a way to calculate that probability; it is not a replacement for clearly identifying the event.
Be Careful About Where “Not” Applies
When a sentence contains more than one event, the position of “not” matters. If the intended event is that \(A\) does not occur while \(B\) does occur, write \(A^c\cap B\). By contrast, “not both \(A\) and \(B\)” means that it is not the case that both occur together. The second phrase includes several possibilities: \(A\) alone, \(B\) alone, or neither event.
Parentheses help show the intended event. Compare \(P(A^c\cap B)\), which means “not \(A\) and \(B\),” with \(P((A\cap B)^c)\), which means “not both \(A\) and \(B\).” The complement is taken around different events, so the results need not be equal.
Worked Example: “Not A and B” Versus “Not Both”
A survey of 90 app users records whether each user has notifications enabled and whether each user uses dark mode. Let \(A\) be the event that notifications are enabled, and let \(B\) be the event that the user uses dark mode. The counts are: 25 users have both features, 25 have notifications enabled but do not use dark mode, 15 use dark mode but do not have notifications enabled, and 25 have neither feature.
Find the probability that a randomly selected user has dark mode enabled but does not have notifications enabled. This wording means “not \(A\) and \(B\),” or \(A^c\cap B\). It describes just the 15 users who use dark mode without notifications.
Now find the probability that the user does not have both features. This means the complement of \(A\cap B\). The 25 users who have both features are excluded, leaving \(90-25=65\) users.
Conclude: The probability of using dark mode but not having notifications enabled is about 0.1667. The probability of not having both features is about 0.7222. They differ because the first event describes one specific group, while the second includes everyone except the users who have both features.
For comparison, “not \(A\)” alone would include everyone who does not have notifications enabled: the 15 users who use dark mode and the 25 who use neither feature. Thus \(P(A^c)=40/90\approx0.4444\). Naming the exact event before using the numbers prevents a count for one phrase from being mistakenly used for another.
A Reliable Notation-Reading Routine
Use this short routine whenever a probability question contains event notation. It is especially useful when “and” and “not” occur in the same sentence.
Write what \(A\) and \(B\) mean in the situation before interpreting a probability expression.
Identify the event described there: \(A\), both \(A\) and \(B\), or not \(A\).
Use the correct region, table cell, or count for that event. For “and,” look for outcomes satisfying both conditions.
Give the result as the chance of the event named in the question, using the relevant sample space.
Common Mistakes and AP Exam Tips
- Treating \(A\) and \(P(A)\) as interchangeable. \(A\) is an event; \(P(A)\) is its probability. A complete answer identifies the event and then gives its probability.
- Reading “and” as “or.” \(P(A\cap B)\) requires both events to occur. In a two-way table, use the cell at the intersection of the two categories, not a row or column total.
- Adding \(P(A)\) and \(P(B)\) to get \(P(A\cap B)\). “And” asks for the overlap, not a sum. Use the count or information that specifically describes outcomes in both events.
- Using “not” on the probability instead of on the event. Write \(P(A^c)\) or \(P(\text{not }A)\) for the probability of the complementary event. Calculate it as \(1-P(A)\) when appropriate.
- Ignoring the scope of “not.” “Not \(A\) and \(B\)” and “not both \(A\) and \(B\)” describe different events. Restate the wording in plain language, and use parentheses in symbols when needed.
- Using the wrong denominator. If a person is selected from the entire group represented in a table, use the grand total. Make sure the numerator counts exactly the event you named.
For full-credit communication, define the events, translate the wording correctly, show a count or probability that matches the event, and state the answer in context. For example: “Let \(A\) be bringing a bottle and \(B\) be buying a snack. The cell for both events contains 30 of the 120 visitors, so \(P(A\cap B)=30/120=0.250\).”
Check Your Understanding
For each question, first state the event in words, then write or interpret the probability notation.
- In a group of 50 people, 18 bring lunch, and 7 of those 18 also bring a drink. If \(L\) means “brings lunch” and \(D\) means “brings a drink,” how many people are counted by \(L\cap D\)?
- If \(P(A)=0.28\), find \(P(A^c)\). State what the result means in words.
- Explain the difference between \(P(A^c\cap B)\) and \(P((A\cap B)^c)\).
- A student writes \(P(A\text{ and }B)=P(A)+P(B)\). Explain why this does not correctly describe the meaning of “and.”
- In a two-way table, where would you look to find \(P(A\cap B)\): a row total, a column total, or the cell where the categories for \(A\) and \(B\) meet?