What Is the Complement of an Event?
In Equally Likely Outcomes and Counting Probability, we calculated an event’s probability by counting the outcomes that meet its definition. Sometimes it is simpler to calculate the probability that an event does not happen. The event “not A” is called the complement of event \(A\).
For example, suppose one product is selected from a batch and \(A\) is the event that it is defective. The complement, written \(A^c\), is the event that the selected product is not defective. These two events describe opposite possibilities for the same selection: either the product is defective, or it is not.
The rule works because \(A\) and \(A^c\) together include every possible outcome in the sample space, and they have no outcomes in common. One of them must occur, and they cannot both occur on the same trial. Thus, their probabilities sum to 1.
The complement depends on how the event is defined. If \(A\) means “a randomly selected product is defective,” then \(A^c\) means “that product is not defective.” It does not automatically mean the product is flawless in every possible way. It means only that it does not meet the stated definition of defective.
Apply the Rule to Defective Products
A product-quality question may give the probability that a product is nondefective and ask for the probability that it is defective, or the other way around. Treat the two descriptions as an event and its complement, using the same product, batch, and selection process for both.
If \(D\) is the event that one randomly selected product is defective, then \(D^c\) is the event that it is not defective. The two probabilities add to 1, so either one can be found by subtracting the other from 1.
Worked Example: Probability a Product Is Not Defective
A factory’s quality model assigns a probability of 0.035 that a randomly selected circuit board from a particular production run is defective. What is the probability that a randomly selected board from that run is not defective?
Let \(D\) be the event that the selected board is defective. Then \(D^c\) is the event that it is not defective. The question gives \(P(D)=0.035\), so use the complement rule:
The probability that the selected circuit board is not defective is 0.965, or 96.5%. The calculation refers to one randomly selected board from the stated production run; it does not promise that exactly 96.5% of every small group of boards will be nondefective.
The subtraction is from 1 because a probability of 1 represents the entire sample space: all possible outcomes of the selection. Subtracting the probability of defective boards leaves the probability of all outcomes that are not defective.
Use Counts to Find the Complement
When outcomes are equally likely, the counting method from the earlier tutorial can be used alongside the complement rule. Count the outcomes in the event or its complement, then divide by the total number of equally likely outcomes. If one count is easier to obtain, the complement rule provides the other probability.
Worked Example: Finding the Defective Share from Inspection Counts
A quality-control team inspects 250 products from a batch. In this invented example, 9 of the inspected products are classified as defective. One of these 250 products is selected at random. What is the probability that it is not defective?
Let \(D\) be the event that the selected product is defective. There are 9 defective products, so \(P(D)=9/250\). The complement is the event that the selected product is not defective. By the complement rule:
The probability that the randomly selected product is not defective is \(241/250\), or 0.964 (96.4%). The count gives the same result directly: \(250-9=241\) products are not classified as defective, so \(241/250=0.964\). The two calculations agree.
The denominator is 250 because the selection is made from those 250 inspected products, each of which is equally likely to be selected. This answer describes a random selection from that set; it is not automatically a probability for products outside the inspected set.
This example also shows how the complement helps check a count. The defective and not-defective counts must add to the total: \(9+241=250\). Their probabilities must also add to 1: \(9/250+241/250=1\).
Find the Probability of Defect from Its Complement
The complement rule works in either direction. If the probability of a product being acceptable under a stated quality definition is known, subtract that probability from 1 to find the probability that it fails to meet that definition. Be precise about what “acceptable” means in the situation.
Worked Example: From Conforming to Nonconforming
For one randomly selected sensor from a production run, the probability that it meets the run’s stated operating specifications is 0.992. Let \(C\) be the event that the sensor conforms to those specifications. What is the probability that it does not conform?
The event “does not conform” is \(C^c\), the complement of \(C\). Apply the rule using the probability given:
The probability that the selected sensor does not conform to the stated operating specifications is 0.008, or 0.8%. This is the probability of failing that defined standard; it should not be described as the probability of every possible kind of defect unless the standard defines nonconformity that way.
A Short Routine for Complement Questions
Before subtracting, check that the two events really are complements. They must refer to the same chance process, the same selection or trial, and opposite outcomes. For example, “the product is defective” and “the product is not defective” are complements when “defective” has a clear definition. “The product is defective” and “the product is scratched” are not necessarily complements, because a product might be neither or both.
State exactly what \(A\) means, including the product or trial and the relevant quality standard.
Write \(A^c\) as “not \(A\)” in the same setting. Check that \(A\) and \(A^c\) cover every possible outcome and do not overlap.
Use the probability that is given: \(P(A^c)=1-P(A)\), or rearrange the rule to find \(P(A)=1-P(A^c)\).
State the resulting probability in context. Confirm that it is between 0 and 1 and that the two complementary probabilities sum to 1.
The complement rule is especially helpful when the probability of one side of a yes-or-no classification is known, or when that side is easier to count. It does not require the outcomes within the event to be equally likely. The key requirement is that \(A\) and \(A^c\) are truly opposite parts of the same sample space.
Common Mistakes and AP Exam Tips
- Subtracting from the wrong number. For a probability, use \(1-P(A)\), not \(100-P(A)\) unless the probability has first been written as a percentage. For example, \(1-0.035=0.965\), while \(100\%-3.5\%=96.5\%\).
- Changing the situation between events. “Defective in this batch” and “not defective in another batch” are not necessarily complements. Keep the selection process, batch, and definition consistent.
- Using a vague complement. Write “not classified as defective under the stated standard,” rather than “perfect,” unless perfection is explicitly defined as the opposite event.
- Assuming any two different descriptions are complements. “Defective” and “scratched” can overlap, and a product might satisfy neither description. A complement must be exactly “not the event,” not merely another event.
- Leaving out context. A complete response says what has the calculated probability. For example: “The probability that a randomly selected board from this run is not defective is 0.965.”
- Forgetting a check. A probability must be between 0 and 1. Also, the probabilities of an event and its complement should add to 1, allowing for rounding if decimals are rounded.
On an AP response, define the event, identify its complement, show the subtraction, and finish with an interpretation in context. If the result is a rounded decimal or percentage, make that clear. For a count-based question, show the total and the relevant count so the reader can see which selection the probability describes.
Check Your Understanding
For each question, define the event and use the complement rule. State your answer in context.
- A production model assigns a probability of 0.024 that one randomly selected gear is defective. What is the probability that it is not defective?
- In a set of 400 inspected packages, 14 are classified as damaged. One package is selected at random from that set. Find the probability that it is not classified as damaged, as a fraction and a decimal.
- The probability that a randomly selected bottle meets a stated fill-volume standard is 0.985. What is the probability that it does not meet the standard?
- Explain why “a product is defective” and “a product is scratched” are not necessarily complementary events.
- A student calculates \(1-0.08=0.92\) and says, “There is a 92% chance the product is perfect.” What wording would more carefully interpret the result if 0.08 is the probability the product fails a stated inspection standard?