Audit the Outcomes Before the Probabilities
In What Makes a Probability Model Appropriate, we described a probability model as a list of possible outcomes paired with probabilities. This tutorial turns that idea into a practical audit for a spinner game. Before calculating the chance of winning a prize, check that the model accounts for every result the game can produce and that the assigned probabilities obey the rules of probability.
A spinner’s physical sections and a game’s outcomes are related, but they are not necessarily the same list. The spinner might have several sections with the same label, while the game records only the prize won. Alternatively, several spinner labels might lead to the same prize. A useful audit starts by tracing each possible spinner result to the game outcome it creates.
These checks answer different questions. The outcome list must be complete and must not double-count a result. Each probability must be within the permitted range. Finally, the total probability must be 1. Passing the arithmetic checks does not guarantee the outcome list is complete: an omitted result can sometimes be hidden by assigning too much probability to another result.
- Identify the result of one play, including what the game records.
- List every distinct game outcome that can occur.
- Check that the outcomes do not overlap and that none are missing.
- Check each assigned probability: \(0\leq P(\text{outcome})\leq1\).
- Add the probabilities and verify that the total is 1.
- Check that the probabilities agree with the spinner description and the game’s rules.
The final check connects the numbers to the situation. For instance, if a spinner has equally likely wedges, the chance of a game outcome depends on how many wedges produce it. A table can satisfy the probability rules and still misrepresent the spinner if it assigns probabilities that do not match the described wedges. As in the earlier tutorial on spinner models, any claim that sections are equally likely depends on the spinner behaving as assumed.
From Spinner Sections to Game Outcomes
Suppose several sections have the same label or pay the same prize. For a model of the prize won, those sections can be combined into one outcome. Add the probabilities of the sections that lead to that prize. The resulting prize categories still need to be mutually exclusive and collectively exhaustive.
This grouping is appropriate only when the question concerns the grouped result. If the question asks which letter the spinner lands on, two sections both labeled “A” do not create different letter outcomes; if it asks which physical wedge was selected, the wedge identities may matter. Always define what one outcome means for the question at hand.
A complete model need not list every path separately if several paths lead to the same recorded result. It does need to account for every path through the game. A useful technique is to make a small mapping table: list each spinner label or section, then record the game result it produces. Group those results only after confirming that no section has been left out.
Worked Example: A Missing Prize on an Eight-Wedge Spinner
Worked Example: A Missing Prize on an Eight-Wedge Spinner
A fictional game uses a spinner with eight equal-sized wedges labeled A, A, B, B, B, C, C, and D. The rules award 0 points for A, 1 point for B, 2 points for C, and 5 points for D. A proposed model lists only 0, 1, and 2 points, with probabilities 0.25, 0.375, and 0.25. Audit the model and give the correct prize model, assuming the eight wedges are equally likely.
State. The chance process is one spin followed by the prize rule. The game outcomes are the point totals, not the letters on the wedges. The possible point outcomes appear to be 0, 1, 2, and 5.
Plan. First compare the proposed point outcomes with the prizes produced by every wedge. Then check the probability bounds and total. Finally, calculate the probabilities from the numbers of equal wedges that award each prize.
Do. The A wedges produce 0 points, the B wedges produce 1 point, the C wedges produce 2 points, and the D wedge produces 5 points. Thus 5 points is missing from the proposed outcome list. Each proposed probability is between 0 and 1, but their total is:
The total is less than 1, which is consistent with the omitted possibility. Counting the wedges for each prize gives the correct probabilities:
Each probability is in the interval from 0 to 1. Their sum is \(0.25+0.375+0.25+0.125=1\). The four prize outcomes are mutually exclusive because a single spin awards only one point total, and collectively exhaustive because every wedge awards one of those totals.
Conclude. The proposed model is incomplete because it leaves out the 5-point prize, and its probabilities add to 0.875 rather than 1. The complete model lists 0, 1, 2, and 5 points with probabilities 0.25, 0.375, 0.25, and 0.125, respectively, under the stated equal-likelihood assumption.
A Total of One Is Not Enough
The eight-wedge example showed an omitted outcome accompanied by a total below 1. But a model can omit an outcome and still have probabilities that add to 1. That is why the outcome-list check and the probability-sum check must be done separately.
For example, if an omitted result is accidentally included in another category’s probability, the listed numbers might sum to 1 even though the categories do not describe the game accurately. A sum of 1 tells us that the assigned numbers satisfy one requirement for a finite probability model. It does not prove that all results are present, that the categories match the rules, or that the probabilities are appropriate for the described spinner.
Also check whether the categories overlap. If a game lists “red,” “blue,” and “red or blue” as three separate outcomes, a red spin fits two listed categories. Those categories are not mutually exclusive as a single-outcome model. The game could instead ask for one event such as “red or blue,” but then that event and its complement should be the two categories in the model.
Worked Example: A Probability Table That Adds to One but Is Wrong
Worked Example: A Probability Table That Adds to One but Is Wrong
A fictional spinner has ten equal wedges: four red, three green, and three blue. A game’s proposed model lists “red” with probability 0.40 and “green” with probability 0.60. The probabilities are between 0 and 1 and add to 1. Is this a valid model of the spinner’s color outcome? Then give two valid ways to describe outcomes for games based on these colors.
State. For the first question, one outcome is the color on one spin. The possible colors are red, green, and blue. The proposed list contains only red and green.
Plan. Check completeness independently of the arithmetic. Then calculate the color probabilities using the ten equal wedges. For alternative models, choose game outcomes that include every color exactly once, either as individual colors or as grouped prize results.
Do. Blue is possible, so the proposed outcome list is not collectively exhaustive. Its probabilities do add to 1: \(0.40+0.60=1\). However, the wedge counts give \(P(\text{green})=3/10=0.30\), not 0.60. The proposed model is therefore not a valid description of the spinner’s color outcome.
One complete model lists each color:
These probabilities are all between 0 and 1 and sum to \(0.40+0.30+0.30=1\). A different game could award a prize for red and a different prize for either green or blue. Its outcomes would be “red” and “not red,” with probabilities \(0.40\) and \(0.30+0.30=0.60\). Those two categories are mutually exclusive and collectively exhaustive for that prize rule.
Conclude. The proposed table is not a valid model of the spinner’s color outcome even though its probabilities add to 1: it omits blue and assigns green the wrong probability. The three-color model is valid for recording color, while the red/not-red model is valid for a game that groups green and blue together.
Check Every Probability and the Total
For a finite model, the probability assigned to an individual outcome cannot be negative or greater than 1. A probability of 0 means the outcome is impossible under the model; a probability of 1 means it is certain. In either case, the outcome still needs to be treated consistently with the game description.
After checking each entry, add all probabilities in the model. If the outcome categories are mutually exclusive and collectively exhaustive, their probabilities must total 1. A total below 1 may signal a missing outcome or an arithmetic error. A total above 1 may signal overlapping categories, an arithmetic error, or incorrect assignments. These clues help locate a problem, but they do not identify it automatically: return to the game description and check the outcome list.
When the spinner has equal-sized wedges and the model assumes each wedge is equally likely, calculate an outcome’s probability by dividing the number of wedges that produce it by the total number of wedges. When wedge sizes are unequal, or the description provides probabilities directly, do not simply count wedges as though they were equally likely. The audit still applies, but the probability source must match the described mechanism.
Worked Example: Finding an Invalid Assignment
Worked Example: Finding an Invalid Assignment
A fictional spinner has ten equal wedges. Four award 0 tokens, three award 1 token, two award 2 tokens, and one awards 5 tokens. A proposed table gives the probabilities 0.40, 0.35, 0.20, and 0.10 for those outcomes, respectively. Audit the table and correct it.
State. The possible game outcomes are 0, 1, 2, and 5 tokens. They are mutually exclusive because one spin awards only one token amount, and they are collectively exhaustive because the rules assign a prize to every wedge.
Plan. Check each proposed probability’s range, add the entries, and compare them with the probabilities implied by the wedge counts. These checks distinguish a probability-range problem from a sum problem or a mismatch with the spinner.
Do. Every proposed entry is between 0 and 1. However, the total is:
The proposed values cannot form a valid probability model because the total exceeds 1. The wedge counts give the corrected values:
Each corrected value is between 0 and 1, and the total is \(0.40+0.30+0.20+0.10=1.00\). The only changed assignment is the probability of 1 token, corrected from 0.35 to 0.30.
Conclude. The proposed table is invalid because its probabilities sum to 1.05. Under the assumption that the ten equal wedges are equally likely, the corrected model assigns probabilities 0.40, 0.30, 0.20, and 0.10 to 0, 1, 2, and 5 tokens, respectively.
Common Mistakes and AP Exam Tips
A strong audit explains both what the model includes and what the numbers do. Avoid treating “the probabilities add to 1” as a complete justification.
- Checking only the sum. A total of 1 does not show that every possible result is listed. Name an omitted result if one exists, as with blue in the ten-wedge example.
- Checking only the outcome names. A complete list can still have probabilities outside the 0-to-1 range or a total other than 1. Report each relevant check.
- Counting labels instead of wedges. If one color appears on several equal wedges, it is not necessarily as likely as a color appearing once. Use the number of wedges that produce the outcome, not just the number of color names.
- Forgetting the game rule. The model may concern prizes rather than spinner labels. Trace every wedge through the rule before deciding which outcomes belong in the model.
- Using overlapping categories. For one play, “red” and “red or blue” cannot be separate mutually exclusive outcomes. Define categories so each play is counted exactly once.
- Assuming a valid arithmetic table must fit the spinner. A table may pass the range and sum checks but assign probabilities that contradict the described wedge counts or game rules.
For full credit, state the possible outcomes in context, explain whether any are missing or overlapping, check the probability bounds and total, and compare the assignments with the spinner description. If the model fails a check, say exactly which requirement fails and give the corrected model when the information allows it.
Key Takeaway
A spinner-game probability model must represent every result the game can produce, with each play belonging to exactly one listed outcome. The probabilities must each be between 0 and 1 and must sum to 1. Finally, those probabilities must agree with the spinner description and the rules that turn a spin into a game result.
Check Your Understanding
For each question, explain which outcome or probability checks apply.
- A six-wedge spinner has two wedges labeled X, three labeled Y, and one labeled Z. A model lists only X and Y. What outcome-list check fails?
- A proposed model lists outcomes A, B, and C with probabilities 0.20, 0.35, and 0.45. What can you conclude from the total, and what must still be checked?
- A game pays 4 points on either a red or a blue wedge and 0 points on a green wedge. What are the distinct point outcomes? Why need they not match the list of colors?
- A probability table assigns 0.55, 0.35, and 0.15 to three listed outcomes. Identify the arithmetic problem.
- A spinner has four equal wedges labeled 1 and one equal wedge labeled 2. Under the assumption that each wedge is equally likely, what probabilities belong in a model of the label spun?