From a Probability Model to Observed Results
A probability model describes what outcomes a chance process can produce and the probabilities assigned to them. In Checking Outcomes and Probabilities in a Model, we considered how to check that the outcome list is complete and the probabilities are valid. Now we can ask a different question: when we repeat the process, do the observed results look reasonably consistent with the model’s predictions?
Imagine rolling a die 120 times. A model that says the die is fair assigns probability \(1/6\) to each face. That does not predict exactly 20 rolls of every face. Instead, 20 is the expected count for each face, while the actual counts can vary from roll to roll. Comparing data with a model means considering both the size of the differences and the variation we could reasonably expect.
An observed relative frequency is a proportion calculated from the data. A model probability is a value the model assigns to an outcome. They are related, but they are not the same thing: the observed proportion describes these particular rolls, while the model probability describes the chance process according to the model.
A Practical Comparison
For each outcome, calculate its observed relative frequency and expected count. Then compare the observed count with the expected count. A useful display puts all three side by side, so a difference is visible without losing sight of how much data were collected.
List the possible outcomes and the probability assigned to each. For a fair six-sided die, each face has probability \(1/6\).
Multiply each model probability by the total number of trials. These are model predictions for the counts, not promises about the results.
Divide each observed count by the total number of trials. Confirm that the counts add to the total and the relative frequencies add to 1, allowing for rounding.
Identify which outcomes are above or below their expected counts and whether the differences seem small or pronounced. Consider how much variation the model allows.
Ask whether the rolls reasonably meet the model’s assumptions. Say whether the results appear consistent with the model; do not treat a mismatch alone as proof that it is false.
For a fair die, the count of one particular face can be described with a binomial model: each roll either shows that face or does not. If the rolls are independent and the face has probability \(1/6\) on each roll, its count among 120 rolls has model \(X\sim B(120,1/6)\). As covered in Mean of a Binomial Distribution and Standard Deviation of a Binomial Distribution, its expected count and standard deviation are:
So, for one specified face under the fair-die model, the count has mean 20 and standard deviation about 4.08 rolls. This gives a scale for judging a difference: being a few rolls away from 20 is not surprising by itself. The six face counts are not independent of one another, however; they must add to 120. The standard deviation for one face is a guide to its individual variation, not a complete measure of how all six counts vary together.
Worked Example: A Fair-Die Model and Modest Differences
Worked Example: A Fair-Die Model and Modest Differences
In a fictional classroom activity, a student rolls a die 120 times and records these counts. Compare the results with the model that the die is fair.
| Face | Observed count | Observed relative frequency | Expected count under fair model |
|---|---|---|---|
| 1 | 18 | 18/120 = 0.1500 | 20 |
| 2 | 22 | 22/120 = 0.1833 | 20 |
| 3 | 19 | 19/120 = 0.1583 | 20 |
| 4 | 21 | 21/120 = 0.1750 | 20 |
| 5 | 17 | 17/120 = 0.1417 | 20 |
| 6 | 23 | 23/120 = 0.1917 | 20 |
State. The claimed model is a fair six-sided die, so each face has probability \(1/6\). There are 120 rolls.
Plan. Calculate the expected count for each face and compare it with the observed count. Also calculate the observed relative frequencies and check that the totals are consistent.
Do. For every face, the expected count is \(120(1/6)=20\). For example, the observed relative frequency for face 1 is \(18/120=0.1500\), while the model probability is \(1/6\approx0.1667\). The observed count for face 1 is 2 below its expected count. Across the six faces, counts total \(18+22+19+21+17+23=120\); the relative frequencies total 1.0000, up to rounding.
The observed counts differ from 20 by \(-2,+2,-1,+1,-3,\) and \(+3\). These are modest differences compared with the standard deviation of about 4.08 rolls for the count of a specified face under the fair model. The data do not show a large discrepancy from that model in this comparison.
Conclude. These 120 rolls appear reasonably consistent with the fair-die model: the observed counts are fairly close to the expected count of 20, with differences on the scale of ordinary variation for an individual face. This comparison does not prove the die is fair; it describes how these results compare with the model.
Worked Example: A Much Higher Count for One Face
Worked Example: A Much Higher Count for One Face
In a separate fictional activity, a die is rolled 120 times. The recorded counts for faces 1 through 6 are 8, 12, 15, 18, 27, and 40. Consider whether these results look consistent with a fair-die model.
State. Under the claimed fair-die model, each face has probability \(1/6\), and each face therefore has expected count 20 among 120 rolls.
Plan. Compare the observed counts with 20 and calculate relative frequencies. For face 6, compare its difference from 20 with the standard deviation for one face’s count under the model.
Do. The counts sum to \(8+12+15+18+27+40=120\), so they account for all the rolls. The observed relative frequency for face 6 is \(40/120=0.3333\), compared with the model probability \(1/6\approx0.1667\). Its count is \(40-20=20\) above the expected count. Relative to the standard deviation of about 4.08, that difference is:
Thus, the count of face 6 is about 4.90 standard deviations above its expected count when considered as the count for that particular face. That is a pronounced difference on the scale of typical variation for one face under the fair model. The other observed relative frequencies are \(8/120=0.0667\), \(12/120=0.1000\), \(15/120=0.1250\), \(18/120=0.1500\), and \(27/120=0.2250\).
This standardized comparison does not calculate the probability of the overall pattern across all six faces. In particular, the six counts are connected because they sum to 120. The calculation helps describe the size of the face-6 difference; it is not, by itself, a complete formal test of the model.
Conclude. The results show a pronounced discrepancy from the fair-die model, especially for face 6. Before concluding that the model is unsuitable for this process, check whether the rolls were recorded accurately and whether the die and rolling method were consistent with the assumptions. These invented results illustrate a comparison, not a real investigation.
Worked Example: Comparing Data with a Nonuniform Model
Worked Example: Comparing Data with a Nonuniform Model
A fictional game designer claims that a particular six-sided die has probabilities 0.12, 0.13, 0.15, 0.18, 0.17, and 0.25 for faces 1 through 6, respectively. In 120 rolls, the observed counts are 14, 17, 16, 23, 19, and 31. Compare these results with the claimed model.
State. The claimed model gives different probabilities to the six faces, so it is not the fair-die model. The probabilities are valid because they add to \(0.12+0.13+0.15+0.18+0.17+0.25=1.00\).
Plan. Multiply each claimed probability by 120 to find the expected count for that face. Then calculate the observed relative frequencies and describe how closely they align with the model predictions.
Do. For face 1, the expected count is \(120(0.12)=14.4\), and the observed relative frequency is \(14/120\approx0.1167\). Applying the same calculations to all faces gives:
| Face | Model probability | Expected count | Observed count | Observed relative frequency |
|---|---|---|---|---|
| 1 | 0.12 | 14.4 | 14 | 0.1167 |
| 2 | 0.13 | 15.6 | 17 | 0.1417 |
| 3 | 0.15 | 18.0 | 16 | 0.1333 |
| 4 | 0.18 | 21.6 | 23 | 0.1917 |
| 5 | 0.17 | 20.4 | 19 | 0.1583 |
| 6 | 0.25 | 30.0 | 31 | 0.2583 |
The expected counts add to 120, as they should: \(14.4+15.6+18.0+21.6+20.4+30.0=120\). The observed counts also add to 120. The observed relative frequencies are generally near the corresponding claimed probabilities; for example, face 6 occurred in \(31/120\approx0.2583\) of the rolls, close to its claimed probability of 0.25.
Conclude. The observed results appear reasonably consistent with the designer’s nonuniform model: the observed relative frequencies are close to the claimed probabilities, and the observed counts are near the expected counts. This comparison does not establish that the probabilities are correct for every future roll.
Common Mistakes and AP Exam Tips
- Calling the expected count a required count. An expected count is a model-based average over repeated use of the process, not a quota each set of rolls must meet. Write “the model predicts an expected count of 20,” not “there must be 20.”
- Comparing counts with probabilities. A count and a probability have different scales. Compare counts with expected counts, or compare observed relative frequencies with model probabilities.
- Assuming every small difference shows a bad model. Outcomes vary by chance even when a model is appropriate. Describe the size of differences and, when useful, compare a count with its model standard deviation.
- Treating a large difference for one outcome as a complete analysis of all outcomes. Face counts are linked because they add to the total. Be clear about whether a calculation concerns one specified face or the full pattern.
- Declaring a model proven by close agreement. A set of results that matches predictions reasonably well does not prove the model true. State that the data appear consistent with the model, given the process and assumptions.
- Ignoring how the data were produced. As discussed in Independence Assumptions in Real Settings, consider whether one roll could affect the next and whether the same die and rolling procedure were used. A numerical comparison is only as meaningful as the model’s fit to the process.
Key Takeaway
Observed relative frequencies describe the results of a particular set of trials; model probabilities describe the outcomes the model assigns. Expected counts connect the two for a stated number of trials. Differences are inevitable, so interpret their size in relation to the model’s expected variation and check whether the process supports the model assumptions.
Check Your Understanding
Use the ideas in this tutorial to compare observed outcomes with claimed model probabilities.
- A die is rolled 120 times, and face 2 appears 24 times. Under a fair-die model, calculate its expected count and observed relative frequency. How do they compare?
- A claimed model assigns probability 0.30 to an outcome in 120 trials. What is its expected count? Explain why that count is not guaranteed to occur exactly.
- For one face of a fair die rolled 120 times, the count has mean 20 and standard deviation about 4.08. In context, describe a count of 24 relative to the mean.
- Why is the count of face 6 not independent of the combined count of the other five faces in 120 die rolls?
- State one careful conclusion to use when observed relative frequencies are close to a model’s probabilities, and explain what that conclusion does not establish.