Why a Probability Table Must Add to One
In Sample Spaces and Outcomes, you learned that a sample space lists every possible complete outcome of a chance process. A probability model assigns a probability to each outcome. The probabilities in a complete table must add to 1 because one of the possible outcomes must occur.
The table must also describe the sample space correctly: its outcomes should cover every possibility, and no two listed outcomes should happen at the same time. If an outcome is missing, or if outcomes overlap, the table may not describe the chance process as intended—even if the numbers happen to add to 1.
These are two separate numerical checks. First, check each probability individually. Second, add all the probabilities. Passing just one check is not enough. For instance, numbers that are all between 0 and 1 might add to 0.92, leaving some probability unaccounted for.
A total of 1 represents certainty that some outcome in the sample space occurs. This is consistent with the complement rule from The Complement Rule: an event and its complement cover all possibilities, so their probabilities add to 1. In a probability table, the same principle applies across all individual outcomes.
The outcomes do not have to be equally likely. For a probability model, what matters is that the assigned probabilities meet the two numerical checks and correspond to the chance process. The equally likely outcomes rule from Equally Likely Outcomes and Counting Probability is useful for particular models, but equal probabilities are not required for every model.
Check the Values and the Total
A reliable check has two parts. Look at every probability to see whether it is at least 0 and at most 1. Then add the probabilities for all outcomes in the table and see whether the sum is 1. If the sum is not 1, the model as written is not valid.
A probability of 0 is allowed: it means the outcome is impossible under the model. A probability of 1 is also allowed: it means the outcome is certain. A negative value or a value greater than 1 is not a probability, even if the table’s total happens to be 1.
Confirm that they cover all possible results and that exactly one listed outcome occurs in a trial.
Every listed probability must be between 0 and 1, inclusive.
Add the values for the full set of outcomes—not just a selection of outcomes. A valid table totals 1.
Say whether the model is valid and identify which check it passes or fails.
Worked Examples: Checking a Probability Model
The examples below use tables to apply both numerical checks. The first table passes; the second shows why a total of 1 alone does not guarantee a valid model.
Worked Example: A Valid Model for a Spinner
An invented spinner has four labeled sectors: red, blue, green, and yellow. A proposed model assigns the following probabilities. Is the model valid?
| Outcome | Red | Blue | Green | Yellow |
|---|---|---|---|---|
| Probability | 0.14 | 0.29 | 0.31 | 0.26 |
Check the outcomes: For this model to describe one spin, the four sectors must be the only possible results, and one spin must land in exactly one sector. The spinner is described as having four sectors, so these outcomes form a complete, nonoverlapping set.
Check each probability: Each value is between 0 and 1, inclusive. No individual value violates the probability range.
Check the total:
Conclusion: The proposed table is a valid probability model for the described spinner. Each outcome has an allowed probability, and the four probabilities add to 1. The unequal values are not a problem: the model does not claim that the sectors are equally likely.
A table can also be used to find the probability of an event. Since an event is a set of outcomes, add the probabilities for the outcomes in that event. For the spinner above, the event “lands on blue or yellow” has probability \(0.29+0.26=0.55\). The total of 1 applies to all possible outcomes, while an event that includes only some outcomes can have a probability less than 1.
Finding a Missing Probability
If a probability table lists every outcome except for one unknown probability, use the requirement that the total is 1. Add the probabilities already given, then subtract that sum from 1. The missing value is the probability needed to complete the model, provided the result is between 0 and 1 and the outcomes really do cover the sample space.
This is an application of the requirement that a complete model totals 1. Do not assume the missing value is equally divided among the listed outcomes: equal probabilities apply only when the model says the outcomes are equally likely.
Worked Example: Finding a Missing Probability for a Delivery Time
In an invented delivery-time model, one delivery is classified as early, on time, or late. The probabilities of an early and an on-time delivery are 0.18 and 0.57. What probability should the model assign to a late delivery?
| Outcome | Early | On time | Late |
|---|---|---|---|
| Probability | 0.18 | 0.57 | \(x\) |
State: Let \(x\) be the probability of a late delivery. The three classifications are intended to include every delivery, and a delivery can belong to only one classification.
Plan: The probabilities for all three outcomes must add to 1. Add the two known probabilities, then subtract their sum from 1 to find \(x\).
Do:
Check: The missing value, 0.25, is between 0 and 1. Adding all three probabilities gives \(0.18+0.57+0.25=1.00\).
Conclude: The model assigns probability 0.25 to a late delivery. Under this model, 25% of deliveries are classified as late in the long run. This completes a valid probability table if the three categories cover all possible delivery times without overlap.
The check is important: solving for a missing value does not by itself prove that the whole model is appropriate. It confirms the arithmetic total and the range of the missing probability. You must still check that the outcomes describe all and only the possible results of the chance process.
When a Table Fails a Check
A proposed table can fail because its probabilities add to a value other than 1, because at least one value is outside the allowed range, or because the outcomes do not correctly describe the sample space. These are different problems, so identify the specific reason rather than saying only that the table “looks wrong.”
Worked Example: A Total of One Is Not Enough
A proposed model for an invented two-outcome device assigns probability \(-0.05\) to “works” and probability \(1.05\) to “does not work.” Is it valid?
Check the total:
The values do add to 1, but the first value is negative and the second is greater than 1. Each outcome probability must individually be between 0 and 1, inclusive. Both values violate that requirement.
Conclusion: This is not a valid probability model. Adding to 1 is necessary, but it is not sufficient; every assigned probability must also be in the allowed range. The table cannot be repaired just by pointing to its total. More information about the device’s chance process would be needed to assign appropriate probabilities.
Now consider a different failure: if every value is in the required range but the sum is 0.96, the table is still invalid as a complete model. The missing 0.04 might represent an omitted outcome, or one or more listed probabilities might be incorrect. Do not automatically assign the difference to a particular outcome unless the question identifies an unknown probability and confirms that the outcome list is complete.
Using the Model to Find an Event Probability
Once a table is established as a valid model, it can answer questions about events made up of several outcomes. Add the probabilities for the outcomes in the event. This works because the outcomes in a probability table are nonoverlapping: a single trial produces one outcome, not two of them at once.
Worked Example: Probability of a Particular Range of Ratings
In an invented rating model, a customer’s rating is recorded as 1, 2, 3, or 4 stars. The model assigns probabilities 0.10, 0.25, 0.40, and 0.25, respectively. Find the probability that a rating is at least 3 stars, and verify that the model is valid.
| Rating | 1 star | 2 stars | 3 stars | 4 stars |
|---|---|---|---|---|
| Probability | 0.10 | 0.25 | 0.40 | 0.25 |
Verify the model: Each rating probability is between 0 and 1. The four possible ratings are distinct and cover the stated rating scale. Their probabilities total:
So the table is a valid model for the stated rating outcomes. The event “at least 3 stars” consists of the outcomes 3 stars and 4 stars. Add the probabilities for just those outcomes:
The model assigns probability 0.65 to a rating of at least 3 stars. This event’s probability is not required to equal 1, because ratings of 1 or 2 stars are possible and are outside the event.
Common Mistakes and AP Exam Tips
- Checking only the sum. A total of 1 does not make a table valid if one of its values is negative or greater than 1. Check every value and the total separately.
- Checking only the individual values. Values such as 0.20, 0.30, and 0.40 are all in range, but they add to 0.90. If these are supposed to be all the outcomes, the model is incomplete or incorrect.
- Forgetting an outcome. Before adding, make sure the table includes every possible outcome. A total of 1 cannot establish that the outcomes actually describe the chance process.
- Assuming outcomes are equally likely. A valid probability model can assign different probabilities to different outcomes. Equal likelihood needs to be part of the model; it does not follow from the fact that outcomes are listed in a table.
- Adding every table value to answer an event question. The total across all outcomes is 1, but a particular event may include only some outcomes. Add only the probabilities for outcomes in the event.
- Giving a missing value without checking it. After calculating \(x=1-S\), check that \(x\) is between 0 and 1 and that including it makes the full total 1.
For a full-credit explanation, state both numerical checks: each listed probability is in the interval from 0 to 1, and the probabilities for all possible outcomes add to 1. When finding a missing probability, show the sum of the known values and the subtraction from 1. Then identify the missing outcome and report its probability in context.
Check Your Understanding
Use the two validity checks and the requirement that the outcomes cover the sample space.
- A model lists three outcomes with probabilities 0.20, 0.35, and 0.45. Are the numerical values consistent with a valid probability model? State both checks.
- A complete table has known probabilities 0.12, 0.33, and 0.29, plus one missing value \(x\). Find \(x\) and verify the total.
- A two-outcome model lists probabilities 0.30 and 0.75. Explain why it is invalid, even though both values are between 0 and 1.
- Can a valid probability model assign probability 0 to an outcome? Explain what that value means in the model.
- A valid four-outcome model has probabilities 0.15, 0.20, 0.25, and 0.40. Find the probability of an event consisting of the first and fourth outcomes.