From Observed Data to a Probability Model
In Reading a Distribution from a Context Description, you learned to connect possible outcomes of a chance process to the values and probabilities of a random variable. Sometimes those probabilities are not given by a die, a spinner, or a written set of game rules. Instead, you have data from repeated observations, summarized in a relative frequency table. You can use the table to construct a probability distribution and treat it as a model for the chance process.
A relative frequency is the proportion of observations in a category. If a category occurs \(f\) times in a data set of \(n\) observations, its relative frequency is \(f/n\). When the categories are the possible values of a discrete random variable \(X\), those relative frequencies can be used as model probabilities \(P(X=x)\). For example, if 12 out of 40 recorded days had a particular value of \(X\), the relative frequency for that value is \(12/40=0.30\).
The relative frequency describes what happened in the data set. The probability in the model represents a chance we use for the process that generated those data. They are closely connected, but they are not identical claims: the observed proportion is known for the recorded data, while the model uses that proportion to represent the process in comparable future repetitions. As in Interpreting a Probability Distribution in Context, a model probability can be interpreted as a long-run proportion if the process and conditions remain comparable.
A Conversion and Checking Process
Start by identifying the random variable and the total number of observations. If the table gives counts, divide each count by that total. If it gives percentages, convert each percentage to a proportion; if it gives relative frequencies already, use those values as probabilities. Keep the values of \(X\) and their probabilities paired in the table.
State what numerical result \(X\) represents for one observation or one repetition of the process.
If the table gives counts, add them to find the number of observations. If it gives proportions or percentages, check what they represent and whether all categories are included.
For a count \(f_x\), calculate \(f_x/n\). For a percentage, divide by 100 to express it as a probability.
Include the relevant possible values, verify that every probability is between 0 and 1 and that the probabilities total 1, then explain what the model represents in context.
The last check is the same validity check introduced in Checking Whether a Probability Distribution Is Valid. A table of counts whose categories exhaust the recorded observations will convert to probabilities that sum to 1, apart from possible rounding. That arithmetic check is important, but it cannot by itself establish that the model is a good description of a future process. The observations should come from a relevant set of comparable repetitions, and the table should represent the values the question asks about.
Worked Example: Minutes Late for a School Shuttle
A school records the number of minutes its shuttle is late on 50 randomly selected school mornings. The invented data are summarized below. Let \(X\) be the number of minutes the shuttle is late on one school morning.
| Minutes late, \(x\) | Number of mornings |
|---|---|
| 0 | 15 |
| 1 | 16 |
| 2 | 11 |
| 3 | 8 |
State. Convert the observed relative frequencies into a probability distribution for \(X\).
Plan. The table gives counts, and their total is 50. Divide each count by 50 to obtain the corresponding model probability. Then check that all listed counts are accounted for and that the probabilities sum to 1.
Do. The total is \(15+16+11+8=50\). The probabilities are calculated as follows:
The probability distribution is:
| Minutes late, \(x\) | \(P(X=x)\) |
|---|---|
| 0 | 0.30 |
| 1 | 0.32 |
| 2 | 0.22 |
| 3 | 0.16 |
Each probability is between 0 and 1, and \(0.30+0.32+0.22+0.16=1.00\). The counts also add to 50, so every recorded morning is represented.
Conclude. According to this data-based model, \(P(X=2)=0.22\): on a randomly selected comparable school morning, the model assigns a 0.22 probability to the shuttle being exactly 2 minutes late. Equivalently, under comparable conditions, about 22% of many repetitions would have \(X=2\). This is a model based on the recorded mornings, not a guarantee that exactly 22% of future mornings will have that result.
Counts, Proportions, and Percentages
Relative frequency tables can present the same information in different forms. Counts need a denominator: use the total number of observations. A proportion such as 0.18 is already on the probability scale. A percentage such as 18% must be written as \(0.18\) to use it as a probability. Check which form the table uses before calculating; dividing a proportion by the total again would make the probability too small.
Use the variable's values as the possible values of \(X\), not the category labels from the table. For example, if \(X\) counts customer requests received in an hour, values 0, 1, and 2 have meaning as values of \(X\). If a table instead groups values into intervals, such as “0–2 requests” and “3–5 requests,” it does not give the probability of each individual value within those intervals. Do not invent that detail.
Worked Example: Battery Alerts in a Device Test
In an invented test, a technician records the number of battery alerts shown during each of 80 device sessions. The relative frequency table reports percentages. Let \(X\) be the number of alerts in one session.
| Alerts, \(x\) | Relative frequency |
|---|---|
| 0 | 55% |
| 1 | 30% |
| 2 | 10% |
| 3 | 5% |
State. Write the table as a probability distribution and interpret \(P(X=1)\).
Plan. The entries are percentages, so divide each by 100. Check that the percentages total 100% and that each listed value of \(X\) is included.
Do. Converting each percentage gives:
The resulting distribution is:
| Alerts, \(x\) | \(P(X=x)\) |
|---|---|
| 0 | 0.55 |
| 1 | 0.30 |
| 2 | 0.10 |
| 3 | 0.05 |
The original percentages total \(55+30+10+5=100\%\), and the probabilities total \(0.55+0.30+0.10+0.05=1.00\). Each probability is between 0 and 1.
Conclude. The model gives \(P(X=1)=0.30\). In context, it assigns a 0.30 probability to exactly one battery alert in a device session. If the tested sessions represent comparable future sessions, the model predicts that about 30% of many such sessions would have exactly one alert.
What Does It Mean to Treat the Table as a Model?
A probability distribution built from data is a simplified description of a chance process. It lets you answer probability questions by using the observed proportions as model probabilities. For instance, once the distribution is set up, a question about \(P(X\geq 2)\) can be answered by adding the probabilities for all listed values at least 2, as in Cumulative Probabilities for Discrete Variables and Probabilities of Ranges and Inequalities.
The model is useful only to the extent that its source data are relevant to the situation being modeled. A table based on weekday observations may not represent weekends if the process behaves differently then. A table from one short period may not capture changes over time. The probability model does not remove those limitations; it makes the assumptions visible. Describe its probabilities as what the model assigns, rather than as certain facts about every future observation.
Also notice that a value absent from the recorded data has relative frequency zero in that data set. That does not always prove the outcome is impossible. If an unobserved value is possible in the real process, a table based only on the observed values may omit it and may not be a suitable complete model. For a probability distribution, include every value the model treats as possible. In a classroom exercise, the table may explicitly define the model's possible values; in a real setting, use context to consider whether unobserved outcomes should be included.
Worked Example: Study-Room Closures at a Library
A library records the number of study rooms closed for maintenance on each of 40 randomly selected days. The invented counts are shown below. Let \(X\) be the number of rooms closed on one day.
| Rooms closed, \(x\) | Number of days |
|---|---|
| 0 | 10 |
| 1 | 14 |
| 2 | 9 |
| 3 | 5 |
| 4 | 2 |
State. Construct a relative-frequency probability model and find the model probability that at least two rooms are closed on a day.
Plan. The total is 40 days. Divide each count by 40 to obtain the probability for that value of \(X\). Then add the probabilities for \(X=2\), \(X=3\), and \(X=4\), since all satisfy \(X\geq 2\).
Do. The total count is \(10+14+9+5+2=40\). Dividing each count by 40 gives:
The probabilities total \(0.25+0.35+0.225+0.125+0.05=1.00\), and each lies between 0 and 1. To find the probability of at least two rooms being closed:
Conclude. The model assigns a probability of 0.400 to at least two rooms being closed on a day. If the selected days reasonably represent comparable days, this means the model predicts that about 40% of many such days would have two or more rooms closed.
Common Mistakes and AP Exam Tips
- Using the wrong denominator. Divide a category count by the total number of observations, not by the number of categories. In the shuttle example, the denominator is 50 mornings.
- Dividing a proportion twice. A relative frequency of 0.30 is already a proportion. Use 0.30 as the probability; do not divide it by the sample size again.
- Leaving percentages on the percentage scale. A probability must be between 0 and 1. Convert 30% to 0.30 before placing it in a probability distribution.
- Mixing up counts and probabilities. A count of 16 mornings is not \(P(X=1)=16\). Divide 16 by the total of 50 to get 0.32.
- Claiming observed results are guaranteed future results. The data show what happened in the recorded observations. The model uses those relative frequencies to describe comparable repetitions; it does not promise the same proportions in a particular future set.
- Ignoring a possible value just because it was not observed. Zero occurrences in a finite data set do not automatically establish that an outcome is impossible. Consider whether the model should include it.
For a clear AP response, define \(X\), show the denominator or percentage conversion, and label the resulting entries as probabilities. Check that the probabilities are between 0 and 1 and total 1. When interpreting, name the event \(X=x\) in context and describe the probability as what the model assigns. If you use it to describe many repetitions, state that the repetitions should occur under comparable conditions.
Key Takeaway
A relative frequency table can be converted into a probability distribution by expressing each category's share of the observations as a probability. The result is a model based on data: it describes the chance process under comparable conditions, while remaining an approximation rather than a guarantee about future observations.
Check Your Understanding
Use each relative frequency table to build or interpret the probability model it suggests.
- In 60 recorded mornings, a garden center sells 0, 1, 2, or 3 seedlings in 12, 21, 18, and 9 mornings, respectively. Define \(X\) as the number of seedlings sold in a morning and construct the probability distribution.
- A table gives the relative frequencies 42%, 38%, 15%, and 5% for \(X=0,1,2,3\). Convert the entries to probabilities and check the total.
- For the table in question 1, find the model probability of selling at least two seedlings in a morning. Interpret it in context.
- A relative frequency table reports a value of 0 for an outcome in the observations. Explain why this alone does not necessarily show that the outcome is impossible.
- Why should an interpretation of a data-based probability model mention comparable conditions?