When a Model Needs Revising
In Limitations of Probability Models, you considered how uncertain inputs or unrealistic assumptions can affect a model’s answer. When you discover a specific flaw, the next step is not to discard every part of the model automatically. Instead, identify what needs to change, revise that part, check that the revised model makes sense, and recalculate the probability of interest.
For example, a fair six-sided die assigns probability \(1/6\) to every face. If evidence about the die’s construction shows that some faces are more likely than others, the equal-probability assumption is not appropriate. The outcomes—1 through 6—may still be complete, but their probabilities must be revised.
A revision should address the flaw you found. If the outcomes are still the right possible results, keep them. If the outcomes are right but not equally likely, change their probabilities. If a process was incorrectly treated as independent, revise the dependence assumption and calculate using the actual process. A revised model is still a model: it does not guarantee what will happen in the next trial.
Check the Revised Probabilities
Before calculating an event probability, confirm that the revised model is valid. Each outcome probability must be between 0 and 1, inclusive, and the probabilities of all mutually exclusive and collectively exhaustive outcomes must sum to 1. These checks, introduced in Checking Outcomes and Probabilities in a Model, still apply after a revision.
Once the model is checked, find the probability of an event by adding the probabilities of the outcomes in that event. This is the same basic calculation used with an equally likely sample space, but the revised outcomes may no longer have equal probabilities.
- Name the specific assumption or probability that the evidence calls into question.
- State what information supports the revised probabilities or conditions.
- Keep outcomes that still represent every possible result, and revise only what needs to change.
- Check that every probability is between 0 and 1 and that the probabilities sum to 1.
- Recalculate the requested event using the revised model, then explain the result in context.
Worked Example: Revising a Fair-Die Model
A game uses a six-sided die. The initial model assumes the die is fair, but its manufacturer confirms that the die is weighted and provides these probabilities for a roll. Let \(X\) be the number shown on one roll. Find the probability that \(X\) is even under the revised model.
| Outcome | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| Probability | 0.10 | 0.10 | 0.15 | 0.15 | 0.20 | 0.30 |
The event is \(X\) is even, which consists of outcomes 2, 4, and 6.
Use the manufacturer’s probabilities rather than the fair-die model’s equal probabilities. Check that the six outcome probabilities are valid and sum to 1.
The probabilities are all between 0 and 1. Their sum is \(0.10+0.10+0.15+0.15+0.20+0.30=1.00\), so they form a valid probability model. Add the probabilities for the even outcomes.
Interpret the sum as the probability of an even result for a roll of this weighted die, assuming the manufacturer’s probabilities apply to the game.
Under the old fair-die model, the probability of an even result would be \(3(1/6)=0.50\). The revised model gives \(0.55\), or 55%. The change follows from the new probabilities: outcome 6, which is even, is more likely than any individual face in the original model.
Conclusion. Based on the manufacturer’s stated probabilities, this weighted die has a 0.55 probability of showing an even number on a roll. That conclusion depends on the supplied probabilities accurately describing the die in the conditions where it is used.
Recalculate for More Than One Trial
A revised probability for one trial can affect probabilities for a sequence of trials. However, do not assume that changing the probabilities also changes every other feature of the process. If rolls of a die are still independent and the die is rolled under the same conditions each time, use the revised probabilities for each roll while retaining the independence assumption. If that assumption is not justified, the model needs another revision.
For an event involving several rolls, identify all combinations of outcomes that satisfy the event. For each combination, multiply the probabilities when the rolls are independent; then add the probabilities of the mutually exclusive combinations. This follows the multiplication and addition rules for probability developed earlier in the course.
Worked Example: A Sum of 7 with the Weighted Die
Use the weighted die from the previous example, with probabilities \(0.10, 0.10, 0.15, 0.15, 0.20,\) and \(0.30\) for outcomes 1 through 6. The die is rolled twice, and each roll is independent of the other. Find the probability that the sum is 7.
Let \(S\) be the sum of the two results. The ordered pairs that give a sum of 7 are \((1,6),(2,5),(3,4),(4,3),(5,2),\) and \((6,1)\). Because the rolls are independent, the probability of each ordered pair is the product of its two outcome probabilities. These pairs are mutually exclusive, so add their probabilities.
Conclusion. Under the revised weighted-die model and the stated independence condition, the probability that the two rolls sum to 7 is \(0.1450\). The fair-die model would give \(6/36\approx0.1667\), so keeping the old probabilities would produce a different answer. The two-roll calculation relies on both the revised probabilities and the assumption that one roll does not affect the other.
When Probabilities Are Estimated from Data
Sometimes the flaw is discovered through observed results rather than information about how a device was designed. In that case, observed relative frequencies can be used as estimates for a revised model, as discussed in Using Data to Check a Probability Model. The relative frequency of an outcome is its observed count divided by the total number of trials.
An estimate based on observed results is not the same as a known long-run probability. A set of trials may show uneven counts because of ordinary chance variation, especially when there are few trials. Treat estimated probabilities as inputs with uncertainty, and avoid claiming that one sample proves a process has a particular probability.
Worked Example: Estimating Revised Die Probabilities
A student rolls a die 200 times to check a model that assumed the die was fair. The results are 18 ones, 22 twos, 28 threes, 32 fours, 44 fives, and 56 sixes. Use the observed relative frequencies as a revised model and find the estimated probability of rolling an even number.
First check the total number of observations: \(18+22+28+32+44+56=200\). Divide each count by 200 to obtain the estimated probability of that face. The resulting probabilities sum to 1 because the six counts include all 200 rolls.
The estimated probability of an even result is the sum of the estimates for 2, 4, and 6. Equivalently, there were \(22+32+56=110\) even results out of 200 rolls.
Conclusion. The observed relative frequency suggests an estimated probability of 0.55 for an even result. This is a possible revised model based on these 200 rolls, not proof that the die’s true probability is exactly 0.55. Further rolls could produce different relative frequencies, and the conditions of the test should resemble the conditions where the die will be used.
Revise Conditions When the Process Is Different
Not every model flaw is fixed by changing a probability value. Sometimes the outcomes and their individual chances look reasonable, but the model describes the wrong process. For example, drawing objects without replacement changes the chance of a later draw because the contents of the container have changed. A model that treats draws as independent with replacement does not describe that process.
To revise a condition, write the probability for the actual sequence of events. After the first draw, update the number of objects of each type and the total number remaining. This makes the changing probabilities explicit rather than treating every draw as if it began with the original contents.
Worked Example: Revising a Model for Draws Without Replacement
A bag contains 4 blue tiles and 3 gold tiles. Two tiles are drawn at random without replacement. An initial calculation incorrectly treated the draws as independent, as if the first tile were returned to the bag. Find the probability that both tiles are blue using the actual process.
Let \(B_1\) be the event that the first tile is blue and \(B_2\) the event that the second tile is blue. The chance of a blue first tile is \(4/7\). Given that the first tile was blue and was not returned, 3 blue tiles remain among 6 total tiles. Thus the chance of a blue second tile, conditional on a blue first tile, is \(3/6\).
If the tiles were replaced after the first draw, the probability would instead be \((4/7)(4/7)=16/49\approx0.3265\). That is not the correct calculation for this situation, because the first blue tile is removed and the composition of the bag changes.
Conclusion. Under the actual no-replacement process, the probability of drawing two blue tiles is \(2/7\), or about \(0.2857\). The model is revised by correcting the condition on the draws, not by pretending the second draw has the same probability as the first.
Common Mistakes and AP Exam Tips
- Keeping equal probabilities after learning outcomes are not equally likely. Use the revised probabilities for the specific outcomes in the event; do not keep using \(1/6\) simply because a die has six faces.
- Changing probabilities without checking their total. A list of plausible-looking values is not a valid probability model unless the probabilities are between 0 and 1 and sum to 1 across all possible outcomes.
- Assuming data frequencies are exact long-run probabilities. Say that the relative frequencies estimate the probabilities. Observed results can vary from one set of trials to another.
- Changing more than the evidence requires. If the die is weighted but repeated rolls remain independent, revise the face probabilities and keep independence if the context supports it. Do not alter an assumption without a reason.
- Ignoring a changed process. When an item is not replaced, the probabilities for later draws change. Use the actual remaining counts rather than reusing the original probabilities.
- Giving a number without naming its conditions. A full-credit explanation identifies the revised model and interprets the result in context, using conditional language such as “under this model” or “assuming the stated probabilities apply.”
Key Takeaway
A useful revision responds to a specific flaw. A weighted die may require new outcome probabilities; observed relative frequencies may provide estimates; and a no-replacement process may require revised conditional probabilities. After changing the model, check it and redo the calculation rather than carrying forward an answer based on the old assumptions.
Check Your Understanding
For each situation, identify the appropriate model revision and explain what should be recalculated.
- A die’s manufacturer reports that its six faces do not have equal probabilities. What must be checked before using those probabilities, and how would you calculate the chance of rolling an odd number?
- A student uses the relative frequencies from 30 rolls as probabilities for future rolls. Why should those values be described as estimates rather than known probabilities?
- A weighted die is rolled twice, and the rolls are independent. What changes from the fair-die model, and what condition can remain unchanged?
- Two tiles are drawn from a bag without replacement. Why might multiplying the first-draw probability by itself give the wrong probability that both tiles are blue?
- A revised model gives probabilities that add to 1. Does that alone establish that the model describes the real process well? Explain.