Evaluating Probability Models — I can critique whether outcomes are equally likely and adjust a probability model when they are not. - Math 2

SECTION 1

Historical Context & Motivation

For centuries, people assumed that games of chance were governed by pure luck or divine will. The idea that you could actually model randomness with mathematics was radical. Early mathematicians discovered that some random events—like rolling a fair die—produce outcomes that are equally likely, while others—like drawing from a loaded deck—do not. Recognizing this distinction became the foundation of modern probability theory and the basis for fields ranging from insurance to medical research.

1654 • The Birth of Probability

Blaise Pascal and Pierre de Fermat exchanged letters about gambling problems, establishing the mathematical rules of probability. They assumed fair dice and fair coins—equally likely outcomes.

1713 • Bernoulli's Ars Conjectandi

Jacob Bernoulli published a landmark work proving the Law of Large Numbers, which showed that experimental frequencies converge to theoretical probabilities over many trials—a tool for testing whether outcomes truly are equally likely.

1900 • Karl Pearson's Chi-Square Test

Karl Pearson developed the chi-square goodness-of-fit test, giving statisticians a formal way to compare observed data against a probability model and decide whether the model fits.

1933 • Kolmogorov's Axioms

Andrey Kolmogorov published a rigorous axiomatic framework for probability. His axioms apply whether outcomes are equally likely or not, making it possible to build models for any random process.

This history reveals a central question that we still face today: How do we know whether a probability model accurately describes reality? If we blindly assume all outcomes are equally likely when they are not, every probability we calculate will be wrong. This lesson teaches you to spot that mistake and fix it.

SECTION 2

Core Principles & Definitions

Before you can critique a probability model, you need to understand what one is and what makes outcomes equally likely versus not equally likely. A probability model is a mathematical description of a random process that lists every possible outcome and assigns a probability to each. These probabilities must satisfy two rules: every probability is between 0 and 1 (inclusive), and the sum of all probabilities equals exactly 1.

Sample Space

The sample space is the complete set of all possible outcomes. For a standard die, the sample space is {1, 2, 3, 4, 5, 6}. Missing even one outcome makes the model incomplete.

Equally Likely Outcomes

Outcomes are equally likely when each has the same probability of occurring. A fair coin has P(H) = P(T) = 0.5. This assumption is valid only when the process is symmetric and unbiased.

Non-Uniform Model

A non-uniform probability model assigns different probabilities to different outcomes. A weighted die might have P(6) = 0.3 while each other face has P = 0.14. Non-uniform models reflect real-world asymmetry.

Theoretical vs. Experimental

Theoretical probability is calculated from assumptions about the process. Experimental (empirical) probability is calculated from observed data. Comparing the two reveals whether the model is accurate.

Relative Frequency

Relative frequency is the ratio of how many times an outcome occurred to the total number of trials. It serves as the basis for building or adjusting a probability model from real data.

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KEY TAKEAWAY

Think of a probability model like a weather forecast. A forecast that predicts "50% chance of rain every single day" is a uniform model—it treats every day the same. But in reality, some days are sunny and some are stormy. A good forecaster adjusts the model based on actual conditions. Similarly, a good probability model matches the data, not just the assumption that everything is fair.

SECTION 3

Visualizing Fair vs. Unfair Models

The diagram below compares a uniform (fair) probability model with observed experimental data for a six-sided die that has been rolled 600 times. If the die were truly fair, each face would appear approximately 100 times. Notice how the observed frequencies differ from the expected ones—this visual gap is what alerts us that the equally-likely assumption may be wrong.

The violet bars show the expected count of 100 per face under a fair (uniform) model. The cyan bars show observed frequencies from 600 actual rolls. Notice that Face 6 appeared 165 times—far more than the expected 100—suggesting the die is loaded.

In the chart above, the fair model predicts each face should appear about 100 times out of 600 rolls. However, Face 6 appeared 165 times—well above the expected value. Meanwhile, Faces 1 and 5 appeared noticeably less. This mismatch is a clear signal that the equally-likely assumption is not valid for this die, and we need to adjust our probability model to reflect the observed data.

SECTION 4

Mathematical Framework

Two key formulas drive the process of evaluating and adjusting probability models. The first is used when you assume equally likely outcomes; the second is used when you adjust the model based on observed data.

UNIFORM (EQUALLY LIKELY) PROBABILITY

P(outcome) = 1 / n

Where n is the total number of equally likely outcomes in the sample space. For a fair six-sided die, P(any face) = 1/6 ≈ 0.1667.

EXPERIMENTAL (RELATIVE FREQUENCY) PROBABILITY

P(outcome) = f / N

Where f is the frequency (count) of the specific outcome and N is the total number of trials. For Face 6 on the loaded die: P(6) = 165/600 = 0.275.

PROBABILITY MODEL VALIDITY CHECK

∑ P(outcomeᵢ) = 1 and 0 ≤ P(outcomeᵢ) ≤ 1 for all i

Any valid probability model—uniform or non-uniform—must satisfy these two conditions. If the probabilities you assign don't sum to 1, or if any probability is negative, the model is invalid.

The process for evaluating a probability model follows a clear logical path. First, identify the sample space and check that it is complete. Second, determine whether the equally-likely assumption is justified—by examining the physical setup, or by collecting data. Third, if the data show that outcomes are not equally likely, compute relative frequencies for each outcome and use those as your adjusted probabilities. Finally, verify that your adjusted model satisfies the validity conditions: all probabilities between 0 and 1, and the total equals 1.

SECTION 5

Adjusting a Probability Model Step by Step

The diagram below illustrates the decision-making flowchart you should follow whenever you encounter a probability scenario. It starts with identifying whether the equally-likely assumption is reasonable and guides you through building or adjusting the model accordingly.

This flowchart shows the decision process: start by identifying the sample space, determine whether outcomes are equally likely (yellow diamond), then either apply the uniform formula (green path) or calculate relative frequencies from data (red/cyan path). Both paths end at the validation check.

Outcome	Observed Frequency (f)	Uniform P(outcome)	Adjusted P(outcome)
Face 1	80	1/6 ≈ 0.167	80/600 ≈ 0.133
Face 2	90	1/6 ≈ 0.167	90/600 = 0.150
Face 3	100	1/6 ≈ 0.167	100/600 ≈ 0.167
Face 4	95	1/6 ≈ 0.167	95/600 ≈ 0.158
Face 5	70	1/6 ≈ 0.167	70/600 ≈ 0.117
Face 6	165	1/6 ≈ 0.167	165/600 = 0.275
Total	600	1.000	1.000

Comparison of uniform vs. adjusted probability model for a loaded die

The table above demonstrates the entire adjustment process. The uniform model assigns each face a probability of approximately 0.167. The adjusted model uses relative frequencies from 600 observed rolls. Notice that the adjusted probability for Face 6 is 0.275—nearly double the fair value. Both models satisfy the validity check: all probabilities are between 0 and 1, and each column sums to exactly 1.

SECTION 6

Worked Example: The Spinner Problem

A carnival game uses a spinner divided into four colored sections: Red, Blue, Green, and Yellow. The game operator claims each color is equally likely. A suspicious player spins the spinner 200 times and records the results: Red = 30, Blue = 85, Green = 45, Yellow = 40. Is the uniform model appropriate? If not, build an adjusted model.

Evaluating and Adjusting the Spinner Model

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Step 1 — Identify the Sample Space

The sample space consists of four outcomes: {Red, Blue, Green, Yellow}. This is complete because the spinner must land on one of these four colors.

S = {Red, Blue, Green, Yellow}, n = 4
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Step 2 — State the Uniform Model

Under the equally-likely assumption, each outcome has probability P = 1/4 = 0.25. This means we'd expect each color to appear about 200 × 0.25 = 50 times out of 200 spins.

P(each color) = 0.25; expected frequency = 50 per color
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Step 3 — Compare Observed Data to Expected

Red appeared 30 times (expected 50), Blue appeared 85 times (expected 50), Green appeared 45 times (expected 50), and Yellow appeared 40 times (expected 50). Blue far exceeds the expected count, and Red falls well below it. These discrepancies suggest the outcomes are not equally likely.

The uniform model does not fit the data—the equally-likely assumption fails.
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Step 4 — Compute Relative Frequencies

Divide each observed frequency by the total number of trials (N = 200): P(Red) = 30/200 = 0.15, P(Blue) = 85/200 = 0.425, P(Green) = 45/200 = 0.225, P(Yellow) = 40/200 = 0.20.

P(Red) = 0.15, P(Blue) = 0.425, P(Green) = 0.225, P(Yellow) = 0.20
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Step 5 — Validate the Adjusted Model

Check that all probabilities are between 0 and 1: ✓. Check that they sum to 1: 0.15 + 0.425 + 0.225 + 0.20 = 1.000 ✓. The adjusted model is valid. Conclusion: the Blue section is likely larger than the others, making it the most probable outcome.

Adjusted model: P(R) = 0.15, P(B) = 0.425, P(G) = 0.225, P(Y) = 0.20. Valid ✓

SECTION 7

Strengths, Limitations, and Common Mistakes

Feature	Uniform Model	Adjusted (Empirical) Model
When to use	Symmetric, unbiased scenarios (fair coin, fair die, well-shuffled deck)	Asymmetric scenarios, unknown biases, or when physical design suggests unequal likelihoods
Strengths	Simple to calculate; no data collection needed; theoretically exact when the assumption is valid	Reflects real-world behavior; self-correcting with more data; applicable to any process
Limitations	Completely wrong when outcomes are not equally likely; cannot be verified without data	Requires sufficient data; small samples may give inaccurate probabilities due to random variation
Accuracy	Perfect if the assumption holds; misleading if it doesn't	Improves as sample size increases (Law of Large Numbers)

Comparing uniform and empirical probability models

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Common Mistake

Students often assume that because there are two possible outcomes (like "pass" or "fail" on a test), each has a 50% probability. But the number of outcomes does not determine whether they are equally likely. Two outcomes can have very different probabilities—think of the chance of being struck by lightning versus not being struck.

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KEY TAKEAWAY

A uniform model is like using a one-size-fits-all shirt. It works great when everyone is roughly the same size, but if the group includes children and adults, the shirt won't fit most people. Similarly, a uniform probability model works only when the underlying process is truly symmetric. When it's not, you need to "tailor" your model to the data using relative frequencies.

SECTION 8

Connection to Advanced Probability & Statistics

Evaluating whether a model fits data is not just a high school skill—it is the backbone of professional statistics. The ideas you are learning now scale up into powerful tools used in medicine, engineering, business, and scientific research.

This Lesson	Advanced Version
Comparing observed frequencies to expected frequencies "by eye"	Chi-square goodness-of-fit test: a formal statistical test that calculates whether the difference between observed and expected is statistically significant
Using relative frequency as estimated probability	Maximum likelihood estimation (MLE): a method that finds the probability model most likely to have produced the observed data
Checking that probabilities sum to 1	Kolmogorov's axioms and probability distributions: formal frameworks that define valid probability measures over any sample space
Deciding if outcomes are equally likely	Hypothesis testing: using p-values and significance levels to rigorously test whether a model should be accepted or rejected

How today's concepts connect to college-level statistics

In AP Statistics and college-level courses, you will learn to quantify how far off observed data must be before you can confidently say a model is wrong. For now, the important takeaway is that comparing models to data is one of the most fundamental practices in all of statistics, and you are building that foundation right now.

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

A bag contains 10 red marbles, 10 blue marbles, and 10 green marbles. If you draw one marble at random, is a uniform probability model appropriate? Explain your reasoning.

Show answer

Yes, a uniform model is appropriate. There are three outcomes (Red, Blue, Green), and each color has the same number of marbles (10 out of 30 total). Since the physical setup is symmetric—each marble is equally likely to be drawn—the equally-likely assumption holds. Each color has a probability of 10/30 = 1/3 ≈ 0.333.

Question 2: BASIC CALCULATION

A thumbtack is flipped 500 times. It lands point-up 320 times and point-down 180 times. Calculate the experimental probability for each outcome and explain why a uniform model would be inappropriate here.

Show answer

P(point-up) = 320/500 = 0.64 and P(point-down) = 180/500 = 0.36. A uniform model would assign P = 0.50 to each outcome, but thumbtacks are not symmetric objects—the shape and weight distribution make one orientation more likely than the other. The data confirm this asymmetry, with point-up occurring nearly twice as often as point-down. The experimental probabilities sum to 0.64 + 0.36 = 1.00, so the adjusted model is valid.

Question 3: INTERMEDIATE

A student surveys 400 classmates about their favorite school lunch: Pizza = 160, Tacos = 100, Salad = 60, Pasta = 80. The student initially assumes each option is equally likely. (a) What probability does the uniform model assign to each option? (b) What adjusted probabilities does the data suggest? (c) Using the adjusted model, what is the probability that a randomly chosen student prefers either Pizza or Tacos?

Show answer

(a) With 4 options, the uniform model gives P = 1/4 = 0.25 for each. (b) Adjusted probabilities: P(Pizza) = 160/400 = 0.40, P(Tacos) = 100/400 = 0.25, P(Salad) = 60/400 = 0.15, P(Pasta) = 80/400 = 0.20. These sum to 1.00. (c) P(Pizza or Tacos) = P(Pizza) + P(Tacos) = 0.40 + 0.25 = 0.65. This means 65% of students prefer one of these two options, which is significantly more than the 50% the uniform model would predict.

Question 4: APPLIED

A quality control inspector tests 1,000 light bulbs from a factory. She categorizes each as Perfect (no defects), Minor Defect, or Major Defect. The results are: Perfect = 870, Minor Defect = 100, Major Defect = 30. (a) Explain why an equally-likely model would be absurd in this context. (b) Build the adjusted probability model. (c) If the factory produces 50,000 bulbs per day, how many would you expect to have major defects?

Show answer

(a) An equally-likely model would assign P = 1/3 ≈ 0.333 to each category, implying that a third of all bulbs have major defects. This is absurd because a functioning factory produces mostly perfect bulbs—if a third had major defects, the factory would shut down. The physical process of manufacturing is designed to produce good bulbs, making the outcomes inherently unequal. (b) Adjusted model: P(Perfect) = 870/1000 = 0.870, P(Minor) = 100/1000 = 0.100, P(Major) = 30/1000 = 0.030. Sum = 1.000 ✓. (c) Expected major defects per day = 50,000 × 0.030 = 1,500 bulbs.

Question 5: CRITICAL THINKING

Two students each roll a die 60 times. Student A gets: {1→12, 2→9, 3→11, 4→8, 5→10, 6→10}. Student B gets: {1→3, 2→5, 3→4, 4→6, 5→7, 6→35}. Both students have the same sample space. (a) For each student, critique whether a uniform model is reasonable. (b) Build an adjusted model for the student whose data does not fit. (c) Discuss how sample size might affect your confidence in rejecting a uniform model.

Show answer

(a) Student A's data: expected frequency is 60/6 = 10 per face. Observed values range from 8 to 12, which is close to 10. Small deviations like this are expected even with a fair die due to random variation. The uniform model appears reasonable for Student A. Student B's data: Face 6 appeared 35 times out of 60 (expected 10), which is 3.5 times the expected count. This extreme deviation strongly suggests the die is loaded, and the uniform model is not appropriate. (b) Adjusted model for Student B: P(1) = 3/60 = 0.050, P(2) = 5/60 ≈ 0.083, P(3) = 4/60 ≈ 0.067, P(4) = 6/60 = 0.100, P(5) = 7/60 ≈ 0.117, P(6) = 35/60 ≈ 0.583. Sum = 1.000 ✓. (c) With only 60 trials, some random variation is expected, so small deviations from a uniform model don't necessarily mean the model is wrong. However, Student B's deviation is so extreme that it is very unlikely to have occurred by chance. With larger sample sizes (hundreds or thousands of trials), we can be more confident when rejecting a model, because random fluctuations become proportionally smaller relative to the expected values.

SUMMARY

Lesson Summary

Summary

A probability model assigns a probability to every outcome in the sample space. When a process is symmetric and unbiased—like a fair coin or a well-shuffled deck—a uniform model assigns each outcome the same probability: P = 1/n. However, many real-world situations involve non-equally-likely outcomes, so assuming a uniform model can lead to completely wrong conclusions.

To critique a model, compare observed frequencies from data against the expected frequencies predicted by the model. If the data and the model disagree significantly, the model should be adjusted. The adjusted model uses relative frequencies (f/N) as the estimated probabilities. Always verify that your final model is valid: every probability must be between 0 and 1, and all probabilities must sum to exactly 1. This skill—evaluating and adjusting probability models—is the foundation of statistical reasoning and data-driven decision making.