Use the Total Probability to Find What Is Missing
In Checking Whether a Probability Distribution Is Valid, you learned to verify that the probabilities in a discrete distribution are between 0 and 1 and add to 1. That same sum-to-one requirement can help when one probability is not provided. If the listed values of a random variable cover every possible outcome, the missing probability is whatever amount brings the total to 1.
A survey often asks each respondent to select exactly one answer from a set of response categories. If those categories include every possible response, they are exhaustive. If a response can fit in only one category, they are mutually exclusive. Together, these features mean each respondent’s answer belongs to exactly one category. The category probabilities therefore account for the whole distribution.
The unknown probability is the remaining share after accounting for the known categories. For example, if the known probabilities sum to 0.76, then the missing probability is \(1-0.76=0.24\). If probabilities are written as percentages, subtract the known percentages from 100% instead.
This method depends on what the survey categories represent. If respondents can select more than one answer, the categories can overlap, and adding their probabilities need not give 1. If there is an unlisted option, the displayed categories may not cover the full set of responses. Before calculating, check the wording: does each respondent choose exactly one answer, and are all possible answers represented?
A Short Plan for Solving
Write the known probabilities in a sum with the unknown probability, set the total equal to 1, and solve. Then check that the result is between 0 and 1, inclusive. Finally, state what that probability represents in the survey. The interpretation should name the response category, rather than stopping at a number.
Confirm the survey assigns each response to exactly one category and includes every possible response.
Add the known probabilities and the unknown probability, and set that sum equal to 1.
Subtract the known total from 1, then verify that the result is an allowed probability.
Explain what share of respondents is represented by the missing category.
Worked Example: Finding a Missing Streaming Preference
A hypothetical survey asks each participant to choose one main way they listen to audio. Let \(X\) be the selected response category. The response options are “music streaming,” “radio,” “downloaded files,” and “other.” Each person must choose exactly one, and “other” includes any answer not named separately. The probabilities for the first three options are 0.28, 0.19, and 0.31. Find the probability of “other.”
| Response category | Probability |
|---|---|
| Music streaming | 0.28 |
| Radio | 0.19 |
| Downloaded files | 0.31 |
| Other | \(x\) |
State. Find \(P(X=\text{other})\), the probability that a randomly selected survey participant chooses “other.”
Plan. Because each participant chooses exactly one response and “other” covers any remaining answer, the categories are mutually exclusive and exhaustive. Their probabilities must sum to 1.
Do. Let \(x=P(X=\text{other})\). Add the known probabilities and the unknown probability, then solve:
The known probabilities add to \(0.78\), so \(x=1-0.78=0.22\). The check \(0\leq0.22\leq1\) confirms that the result is within the allowed range. The total is \(0.28+0.19+0.31+0.22=1.00\).
Conclude. The probability that a randomly selected participant chooses “other” is 0.22, or 22%. This is the remaining share after the other three response categories are accounted for.
Working with Percentages
Survey summaries often report percentages rather than probabilities. The same reasoning applies: if the response options are mutually exclusive and exhaustive, their percentages must total 100%. To find a missing percentage, subtract the known percentages from 100%. If you need to give the result as a probability, divide the percentage by 100.
Keep the units consistent while calculating. Do not subtract a decimal probability such as 0.35 directly from 100%; use either all decimals with a total of 1 or all percentages with a total of 100%. A final conversion is straightforward: 17% is a probability of 0.17.
Worked Example: A Missing Community Workshop Choice
In a hypothetical community survey, each person is asked to select the one type of weekend workshop they would be most likely to attend. The options are gardening, cooking, bicycle repair, and art; the survey includes every possible choice. The reported percentages are 26% for gardening, 18% for cooking, 34% for bicycle repair, and an unknown percentage for art. Find the probability of choosing art.
State. Find the proportion of respondents who would choose the art workshop.
Plan. Since each respondent chooses one option from the complete list, the percentages must total 100%. First find the missing percentage; then write it as a probability.
Do. The known percentages total \(26+18+34=78\)%. The missing percentage is \(100-78=22\)%. Converting to a probability gives \(22/100=0.22\).
Conclude. The probability that a randomly selected respondent chooses the art workshop is 0.22, or 22%. As a check, \(26\%+18\%+34\%+22\%=100\%\).
The answer is a probability for a randomly selected respondent under the survey’s described distribution. It does not say that exactly 22% of every future group will choose art. The task here is to complete the probability distribution, not to make a claim about a different population or future survey.
Check What the Missing Value Implies
A subtraction can always produce a number, but that number is not automatically a valid probability. If the known probabilities already add to more than 1, subtracting their sum from 1 gives a negative result. That signals a problem: the values cannot all be probabilities in a distribution as stated. Similarly, a result greater than 1 cannot be a missing probability. Use the range check as well as the sum.
A result of 0 is possible: it means the model assigns no probability to that response category. A result of 1 is also possible, but it leaves no probability for any other category. These boundary cases follow the same inclusive range requirement discussed in the earlier tutorial on checking distributions.
Worked Example: A Missing School-Event Preference
A hypothetical student survey asks each respondent to select exactly one preferred kind of school community event from a complete list: a concert, a board-game evening, a food fair, a film night, or a sports event. The probabilities given for the first four categories are 0.14, 0.22, 0.27, and 0.18. Find the probability of choosing a sports event and explain the result in context.
State. Find \(P(X=\text{sports event})\), where \(X\) is the respondent’s selected event preference.
Plan. The survey requires one choice, and the list includes every possible response. Thus, the category probabilities sum to 1. Add the known values, subtract from 1, and check the result’s range.
Do. The known probabilities total \(0.14+0.22+0.27+0.18=0.81\). Therefore, the missing probability is \(1-0.81=0.19\). It is between 0 and 1, and adding it back gives \(0.81+0.19=1.00\).
Conclude. The probability that a randomly selected respondent chooses the sports event is 0.19, or 19%. The response categories together account for the full probability of 1.00.
The context check matters here. The calculation works because the question specifies one choice from a complete list. If respondents could choose a concert and a food fair, for example, the events would overlap. Their probabilities would not necessarily add to 1, so the missing value could not be found by this subtraction alone.
Common Mistakes and AP Exam Tips
- Subtracting from the wrong total. Probabilities use a total of 1; percentages use a total of 100%. Keep the format consistent throughout the calculation.
- Forgetting the response categories must cover the whole distribution. State that the categories are mutually exclusive and exhaustive, or point to the survey wording that makes this clear. If choices can overlap or options are omitted, the sum-to-one setup may not apply to the listed categories.
- Stopping after the arithmetic. A numerical result should be checked against the probability range. If it is negative or greater than 1, explain that the given values do not produce a possible missing probability.
- Giving no contextual interpretation. A full-credit conclusion identifies the category and the group or process, such as “The probability that a randomly selected respondent chooses the art workshop is 0.22.”
- Treating a probability as a guaranteed sample percentage. A probability describes the distribution in the model; it does not guarantee an exact proportion in every group of respondents.
A clear response shows the equation, the known total, and the subtraction. For example: “The response categories are mutually exclusive and exhaustive, so their probabilities sum to 1. The known probabilities sum to 0.81; therefore, the missing probability is \(1-0.81=0.19\). Thus, 19% of respondents are represented by the missing category in this distribution.” This explains both why the calculation is valid and what the result means.
Key Takeaway
When a survey distribution has one missing category probability, use the total probability of 1 as a whole. Subtract the sum of the known probabilities, check that the result lies between 0 and 1, and interpret it as the probability of the missing response.
Check Your Understanding
Assume each survey respondent selects exactly one answer from a complete list. For each question, show the calculation and interpret the missing probability in context.
- A survey about preferred park activities lists walking at 0.36, birdwatching at 0.21, and picnicking at 0.18. What is the probability of selecting the remaining activity category?
- A survey reports that 32% of respondents prefer a morning class and 29% prefer an afternoon class. The only other option is evening. Find the probability of preferring an evening class.
- Known probabilities for three response categories are 0.40, 0.35, and 0.30. What value would the sum-to-one calculation give for a fourth category? Can that be a valid missing probability? Explain.
- Can the missing probability equal 0? Explain what that would mean for the response category.
- Why might the sum-to-one method fail if survey participants can select more than one response?