Look Beneath the Probability Numbers
In What Makes a Probability Model Appropriate and Checking Outcomes and Probabilities in a Model, we checked whether a model includes the relevant outcomes and assigns probabilities that obey the rules of probability. There is another question to ask: What must be true about the chance process for those particular probabilities to make sense?
A model may call a die “fair,” describe a selection as “random,” or treat repeated results as independent without explaining what those words assume. Those assumptions matter. If a die is loaded, if a selection favors some people, or if an earlier result changes what can happen next, calculations based on a different process may not describe the situation.
An assumption is not automatically unreasonable just because it is unstated. Some are supported by the way a device or process is designed. Others are simplifications made to calculate probabilities. The important step is to identify the assumption, connect it to the process, and avoid presenting a model-based result as guaranteed fact.
Three Assumptions to Check
A model using a fair six-sided die assumes that each side has probability \(1/6\) on a roll. “Fair” is a claim about the probabilities, not merely a description of the die’s appearance. A die can look balanced and still be weighted or behave differently when rolled in a particular way.
A statement that a choice is random makes a claim about how the selection is made. It does not, by itself, mean every possible result has the same probability. A process can randomly select among choices while assigning them unequal chances. Conversely, calling a process random does not establish that it is actually free from patterns or preferences.
Independence is a claim about how results relate across events or trials. Two events are independent when the occurrence of one does not change the probability of the other. For independent events \(A\) and \(B\), the probability that both occur is the product of their individual probabilities:
For repeated trials, independence means an earlier result does not alter the chance of a later result. It does not mean the results are identical or that they must be different. The claim needs to fit the mechanism: for example, whether the process resets, whether items are replaced, or whether conditions change.
- Equal likelihood: Does the model assume each listed outcome has the same probability? What feature of the process supports that claim?
- Randomness: How is the result selected or generated? Could the method systematically favor some outcomes?
- Independence: Could one result change the probabilities for later results? Does the process reset or otherwise keep the chances stable?
- Scope: Does the assumption apply to the exact situation being modeled, or only to a simplified version of it?
These questions are related but not interchangeable. Equal likelihood concerns the probabilities of outcomes in a particular trial. Independence concerns whether one event changes the probability of another. A process can have equally likely outcomes on each trial but still produce dependent results, or it can have independent results whose possible outcomes are not equally likely.
Worked Example: What “Fair Die” Adds to a Model
Worked Example: What “Fair Die” Adds to a Model
A game uses a six-sided die and awards a bonus if two consecutive rolls are both sixes. The game description does not say whether the die is fair or whether the rolls are independent. Find the probability of the bonus under those assumptions, and explain what the result depends on.
State. Let \(A\) be the event that the first roll is a six, and \(B\) the event that the second roll is a six. The bonus event is \(A\) and \(B\).
Plan. A calculation using \(1/6\) for each six requires equally likely sides on each roll. Multiplying the two probabilities also requires independence. State these assumptions rather than treating them as facts supplied by the phrase “six-sided die.”
Do. If the die is fair, then \(P(A)=P(B)=1/6\). If the rolls are also independent, then:
The same result can be checked by counting ordered pairs of rolls. There are \(6\cdot6=36\) equally likely pairs under the assumptions, and exactly one pair, \((6,6)\), gives the bonus. Thus the probability is \(1/36\), or about 0.0278 rounded.
If the die is not fair, the probability of a six may not be \(1/6\). For example, suppose a particular die has probability 0.20 of landing on six on each roll. If the rolls are independent, the bonus probability is \(0.20\cdot0.20=0.0400\), not \(1/36\). If the die’s behavior changes between rolls, even that calculation may not apply.
Conclude. The probability \(1/36\) is justified only under both the fair-die assumption and the independence assumption. The description “six-sided die” alone does not establish either one.
Random Does Not Mean Equally Likely
In everyday speech, “random” is sometimes used as though it meant “every outcome has the same chance.” In a probability model, keep the ideas separate. Randomness describes the chance-based selection process; equal likelihood is a specific probability claim that needs support.
A computer program could randomly choose an option while assigning different selection probabilities to different options. A randomly drawn entry from a container could also have unequal chances if some entries appear more often than others. To evaluate the model, ask how the chance mechanism works rather than relying on the word “random” alone.
Worked Example: A Random App With Unequal Chances
A fictional lunch-planning app randomly selects one of four suggestions. Its documented selection settings assign probabilities 0.40 to soup, 0.30 to a sandwich, 0.20 to a salad, and 0.10 to pasta. A student argues that, because the choice is random, each suggestion must have probability 0.25. Evaluate the claim and find the probability that the app suggests soup or a sandwich.
State. The outcomes are the four lunch suggestions. The stated model says the app makes a random selection but uses unequal selection probabilities.
Plan. Check whether the stated probabilities form a valid model, then use the model’s assigned probabilities for the requested combined event. Do not replace the stated probabilities with equal chances merely because the process is random.
Do. Each probability is between 0 and 1, and their sum is:
The model is consistent with unequal chances. The probability of soup or a sandwich is the sum of their probabilities because a single app selection cannot be both suggestions:
As a check, the two remaining suggestions have total probability \(0.20+0.10=0.30\), and \(0.70+0.30=1.00\). If all four suggestions were equally likely, each would instead have probability \(1/4=0.25\), but that would be a different model and would contradict the documented settings.
Conclude. A random selection need not give each outcome the same chance. Under the app’s stated model, the probability of soup or a sandwich is 0.70; assigning 0.25 to every suggestion would not match the described selection process.
Independence Depends on the Process
A common hidden assumption is that repeated results are independent. It can be tempting to use the same probability for each result and multiply, but that step needs justification. Ask what happens between results: Are items returned? Does a device reset? Could earlier outcomes change the conditions or the available options?
As discussed in Spotting Non-Binomial Situations, drawing without replacement can make later probabilities depend on earlier draws. The same reasoning helps audit a probability model, even when the task is not to decide whether a binomial model applies. In contrast, replacing a selected item before the next draw may restore the original composition, though other features of the process could still matter.
Worked Example: Two Raffle Winners From One Class
A fictional school raffle has 24 tickets, one for each student. Eight tickets belong to students in the robotics club. Two different tickets are drawn without replacement for two prizes. Find the probability that both winners are in the robotics club, and compare it with the result from incorrectly treating the draws as independent.
State. Let \(A\) be the event that the first winner is in the robotics club and \(B\) the event that the second winner is in the club. There are 8 club tickets and 24 tickets in total.
Plan. Because the first ticket is not replaced, the number of club tickets and total tickets remaining depend on the first result. Calculate the chance of a club winner first, then the conditional chance of a club winner second given that the first winner was in the club. Compare this with the product of the original probability twice, which would assume independence.
Do. The probability of a club winner first is \(8/24\). Given that the first winner is from the club, 7 of the 23 remaining tickets belong to club members. Therefore:
The fraction can also be checked by reducing \(56/552\): dividing numerator and denominator by 8 gives \(7/69\). If the draws were incorrectly treated as independent, the calculation would be \((8/24)^2=(1/3)^2=1/9\approx0.1111\). That is not the correct probability for this raffle because the first ticket is not replaced and the second chance changes after a club ticket is drawn.
Conclude. The probability that both raffle winners are in the robotics club is \(7/69\), or about 0.1014. The independent-draw calculation gives a different value because the raffle’s no-replacement rule makes the two results dependent.
Assumptions Can Be Plausible Without Being Certain
A probability model describes a chance process under specified conditions; it does not prove those conditions are true. A physical die may be designed to be balanced, or an app may be programmed with specified selection settings. Those details support assumptions, but a model should still be matched to how the process is actually used.
For example, a die rolled on an uneven surface may not behave like the same die rolled on a level table. A process that usually has stable conditions may change when a device is damaged or a procedure is altered. The question is not whether an assumption is perfectly certain. It is whether there is a reasonable basis for using it for the situation at hand, and whether a different assumption would materially change the conclusion.
When the information is incomplete, say so. A careful response might state, “If the die is fair and successive rolls are independent, the probability is \(1/36\). The description does not provide enough information to verify those assumptions.” That is more accurate than claiming the probability must be \(1/36\) simply because the object has six sides.
Worked Example: Does a Made Shot Change the Next Chance?
Worked Example: Does a Made Shot Change the Next Chance?
In a fictional practice drill, a player makes the first shot with probability 0.70. After making the first shot, the player makes the second with probability 0.80, perhaps because the first success builds confidence. Find the probability that both shots are made. Compare it with the result from an independence assumption using the 0.70 probability for both shots.
State. Let \(A\) be the event that the first shot is made and \(B\) the event that the second shot is made. The stated conditional probability is \(P(B\mid A)=0.80\), while \(P(A)=0.70\).
Plan. Use the first-shot probability and the second-shot probability given that the first shot was made. Then compare with the product \(0.70\cdot0.70\), which assumes that the chance of the second make is unchanged after a first make.
Do. The chance of making both shots is:
If the shots were independent and each had make probability 0.70, the model would give \(0.70\cdot0.70=0.49\). The two results differ by \(0.56-0.49=0.07\). The calculation using 0.49 is not supported by the stated condition that the second-shot chance after a first make is 0.80.
Conclude. Under the stated probabilities, the chance of making both shots is 0.56. Treating the shots as independent would give 0.49, but that assumption conflicts with the higher stated second-shot probability following a first make.
Common Mistakes and AP Exam Tips
A strong explanation does more than name an assumption. It identifies what the assumption says about the process and what follows if it does not hold.
- Equating “six-sided” with “fair.” A die can have six faces without having six equally likely results. Full-credit wording identifies fairness as an assumption about the chance of each face.
- Equating “random” with “equally likely.” Random selection can use unequal probabilities. State what the selection mechanism implies rather than assigning equal chances automatically.
- Multiplying without checking independence. A product such as \(P(A)P(B)\) for the chance of both events requires independence. If the second probability depends on the first outcome, use the appropriate conditional probability instead.
- Assuming the same probability on every try. Repeated trials may not have stable chances if the available items or conditions change. Explain what changes, or why the process is treated as stable.
- Presenting a model result as a known fact. Use qualifying language such as “assuming the die is fair” or “under the independence assumption” when those details are not established.
For full credit, name the assumption, connect it to the model’s probability assignment or calculation, and explain what information would support it or how the result could change if it failed. If the scenario does not establish an assumption, do not invent evidence for it.
Key Takeaway
Probability calculations depend on more than a complete outcome list and valid arithmetic. A model may also rely on fairness, randomness, or independence. Make those assumptions visible, check them against the described chance process, and qualify the result when the information does not establish them.
Check Your Understanding
For each question, identify the assumption involved and explain what must be checked.
- A game uses a six-sided die and assigns probability \(1/6\) to each side. What assumption does this make about the die?
- A website says it “randomly” assigns one of three promotional offers, but gives no information about its selection settings. Can you conclude that each offer has probability \(1/3\)? Explain.
- Two tickets are drawn from a box without replacement. Why might the probability of a particular result on the second draw depend on the first?
- A student calculates the probability of two events happening by multiplying their separate probabilities. What assumption must be justified?
- A model predicts a probability under a stated fairness assumption, but the description does not say whether the device is fair. How should the result be communicated?