Bayes Rule Playground | AI Playgrounds

Scenario:

P(disease) 1.0%

How common is the condition in the population?

P(test+ | disease) 99.0%

Sensitivity: when sick, % the test correctly says so.

P(test− | healthy) 95.0%

Specificity: when healthy, % the test correctly says so.

Population shown 10,000

Larger = smoother visualization; same math regardless.

P(disease | test+) = [P(test+ | disease) × P(disease)] ÷ P(test+)

Likelihood ratios: LR⁺ = sensitivity ÷ (1 − specificity) = - LR⁻ = (1 − sensitivity) ÷ specificity = -

LR⁺ > 10 = strong positive evidence. LR⁻ < 0.1 = strong negative evidence. Multiply the prior odds by the LR to get posterior odds (the way doctors actually compute Bayes).

Run a second test: The posterior from the first test becomes the prior for the second. Watch P(sick) update twice.

Has disease (red) Healthy (green) Yellow ring = test + Blue ring = test −

If your test is POSITIVE, prob. you're actually sick:

If your test is NEGATIVE, prob. you're actually sick:

🌳 Show probability tree (alternative representation)

🎯 Try these scenarios first

Pick a scenario, predict what should happen, then apply it and compare the result with your prediction.

Scenario 1: A rare condition with a strong test

Core question: Why can a 99 percent accurate test produce only about a 50 percent posterior probability?

Run and watch: Use a 1 percent base rate with 99 percent sensitivity and 99 percent specificity, then inspect the people counts after a positive result.

Predict first: Before applying, estimate the posterior probability and identify which group will create most positive results.

Explain afterward: Explain how the large healthy population creates enough false positives to compete with true positives.

Scenario 2: A small loss of specificity has a large effect

Core question: Why does specificity matter so much when the condition is rare?

Run and watch: Keep the 1 percent base rate and 99 percent sensitivity, but lower specificity to 95 percent.

Predict first: Predict whether the posterior will fall slightly or dramatically.

Explain afterward: Use the false-positive count to explain the size of the change.

Scenario 3: A higher base rate changes the meaning of the same test

Core question: How can the same test result mean something different in a higher-risk population?

Run and watch: Raise the prior probability to 20 percent while keeping the test settings fixed.

Predict first: Predict how much the posterior will increase before you apply the scenario.

Explain afterward: Explain why changing the population changes the interpretation even though the test did not change.

Scenario 4: A second independent positive result

Core question: Why can repeated evidence change belief more strongly than one result?

Run and watch: Start with the rare-condition scenario, apply one positive result, then use the positive-test button again.

Predict first: Predict whether the second positive adds the same number of percentage points as the first.

Explain afterward: Explain how the first posterior becomes the prior for the second update.

Scenario 5: A spam filter uses the same mathematics

Core question: What changes when false positives and false negatives have different consequences?

Run and watch: Switch to the spam-filter context and compare a common event with the rare-condition case.

Predict first: Predict which error type will be more costly for a real inbox.

Explain afterward: Separate the probability calculation from the decision about which error matters more.

✨ Featured comparison

Featured comparison

Change the base rate before touching test accuracy. A strong test can still produce a surprising posterior when the condition is rare.

Run one scenario with a rare condition and one with a common condition, then compare posterior probability.

🧭 Visual explanation

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📚 Lesson tour (5 steps)

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📝 Worksheet (3 questions)

Q1. Disease test. Set prior=1%, sensitivity=99%, specificity=99%. P(disease | positive) is closest to:

~99%: the test is 99% accurate, so a positive is almost certainly real. ~50%: false positives from the huge healthy majority roughly equal the true positives. ~1%: one test barely moves the 1% prior at all.

Q2. Hold sens=99%, spec=99%. Raise prior from 1% to 20%. P(disease | positive) goes:

Down: a higher prior leaves less room for the posterior to rise. Same: the test's accuracy is unchanged, so the result is unchanged. Up: a higher prior means more true positives relative to false positives.

Q3. Back to prior=1%, sens=99%, spec=99%. After ONE positive (posterior ≈ 50%), click "Test came back POSITIVE" to retest. After the SECOND positive, P(disease) is closest to:

~99%: the second positive updates from the new ~50% prior and multiplies the odds. ~50%: a repeat of the same test just gives the same result again. ~90%: two positives add up but never get near certainty.

🧪 Student response packet

Predict how a positive result changes belief, adjust the prior and test accuracy, then explain the base-rate effect.

Predict

Observe

Explain

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↓ Read the explanation

Imagine a condition that affects 1 person in 100. A test has 99% sensitivity and 99% specificity: it detects 99% of people who have the condition and correctly clears 99% of people who do not. Your result is positive. The probability that you actually have the condition is not 99%. With these numbers, it is about 50%. Bayes' rule explains why.

Start with the four quantities

Prior or base rate: how common the condition is before seeing the test.
Sensitivity: P(positive | condition), the true positive rate.
Specificity: P(negative | no condition), the true negative rate.
Posterior: P(condition | positive), the probability you want after seeing the result.

Sensitivity and specificity describe the test after the true condition is known. The posterior reverses the conditioning: it asks about the condition after the test result is known. Bayes' rule performs that reversal by including the prior.

Why the answer is about 50%

Count 10,000 people. With a 1% prior, 100 have the condition and 9,900 do not. At 99% sensitivity, 99 of the 100 affected people test positive. At 99% specificity, 1% of the 9,900 unaffected people also test positive, which is 99 false positives. The positive group therefore contains 99 true positives and 99 false positives. The posterior is 99 / (99 + 99) = 0.50.

Bayes' rule can be understood as group counting. A small false positive rate applied to a very large unaffected group can produce as many positive results as a high sensitivity applied to a small affected group.

The two main levers

Two controls have especially strong effects on the posterior:

The prior. Make the condition rarer while holding test quality fixed. The false positives come from a much larger group, so the posterior usually falls.
The specificity. Hold the prior fixed and increase specificity. The false positive rate, 1 − specificity, shrinks, often raising the posterior sharply for rare conditions.

The spam scenario uses the same structure

Select the spam-filter scenario. Spam is usually more common than a rare medical condition, so the prior changes. The costs also change: a false positive sends a legitimate message to junk, while a false negative lets spam through. The probability calculation has the same form even though the interpretation and decision costs differ.

What this model leaves out

Real diagnostic tests often produce continuous measurements rather than a single positive or negative label. Changing the decision threshold trades sensitivity against specificity. Multiple test results may also be dependent, so they cannot always be multiplied as though each were new, independent evidence. Real spam filters combine many features. The applet isolates the base-rate mechanism before those complications are added.

For teachers

Curriculum: Conditional probability and Bayes' rule. Distinguish P(test | condition) from P(condition | test), interpret a confusion matrix, and explain why base rates matter.

Pre-exploration prompts:

If a test is 99% sensitive and 99% specific, what else must you know before interpreting a positive result?
How are sensitivity and specificity different from the probability that one positive result is correct?
Why can a low false positive rate still create many false alarms?

Post-exploration prompts:

Set prior, sensitivity, and specificity to 1%, 99%, and 99%. Reconstruct the result with 10,000 people.
Lower the prior to 0.1% while keeping test quality fixed. Which count changes most?
Hold the prior at 1% and raise specificity. Why is this control so influential?
Switch to spam filtering. Which probability structure stays the same, and which decision costs change?

Misconceptions to test: test accuracy equals the posterior; a positive result means the condition is probably present; Bayes' rule is detached from ordinary counting.

设想某种情况在 100 人中影响 1 人。某项检测的灵敏度和特异度都为 99%：它能检出 99% 的患病者，也能正确排除 99% 的未患病者。你的检测结果为阳性。你真正患有该情况的概率并不是 99%。按照这些数值，答案约为 50%。贝叶斯定理解释了原因。

先明确四个量

先验概率或基率：在看到检测结果之前，该情况有多常见。
灵敏度：P(阳性 | 患病)，也就是真阳性率。
特异度：P(阴性 | 未患病)，也就是真阴性率。
后验概率：P(患病 | 阳性)，即看到结果后真正想求的概率。

灵敏度和特异度描述的是在真实情况已知时检测如何表现。后验概率把条件反过来：在检测结果已知时，真实情况有多可能。贝叶斯定理通过加入先验概率完成这种条件转换。

为什么答案约为 50%

以 10,000 人为例。先验概率为 1%，因此 100 人患病，9,900 人未患病。灵敏度为 99%，所以 100 名患病者中有 99 人检测为阳性。特异度为 99%，意味着 9,900 名未患病者中有 1% 也会检测为阳性，即 99 个假阳性。因此阳性人群中有 99 个真阳性和 99 个假阳性。后验概率为 99 / (99 + 99) = 0.50。

贝叶斯定理可以理解为对不同人群进行计数。即使假阳性率很低，只要它作用于一个非常大的未患病人群，产生的阳性数量也可能与小型患病人群中的真阳性一样多。

两个主要杠杆

下面两个控制量对后验概率尤其重要：

先验概率。保持检测质量不变，让该情况变得更罕见。假阳性来自更大的人群，因此后验概率通常会下降。
特异度。保持先验概率不变并提高特异度。假阳性率 1 − 特异度 会降低，这通常会显著提高罕见情况的后验概率。

垃圾邮件情境使用相同结构

选择垃圾邮件过滤情境。垃圾邮件通常比罕见疾病更常见，因此先验概率不同。错误的代价也不同：假阳性会把正常邮件送入垃圾箱，假阴性则会让垃圾邮件通过。概率计算的结构相同，解释和决策成本却不同。

这个模型省略了什么

真实诊断检测通常产生连续测量值，而不是单一的阳性或阴性标签。改变判断阈值会在灵敏度和特异度之间形成权衡。多次检测结果也可能相互依赖，因此不能总把它们当作独立的新证据直接相乘。真实垃圾邮件过滤器会组合大量特征。本工具先单独呈现基率机制，再让学习者理解这些更复杂的情况。

教师指导

课程目标：条件概率与贝叶斯定理。区分 P(检测 | 真实情况) 与 P(真实情况 | 检测)，解释混淆矩阵，并说明基率为什么重要。

探索前问题：

如果某项检测的灵敏度和特异度都是 99%，在解释阳性结果前还必须知道什么？
灵敏度和特异度与某一次阳性结果正确的概率有什么不同？
为什么很低的假阳性率仍可能产生许多误报？

探索后问题：

把先验概率、灵敏度和特异度设为 1%、99% 和 99%，用 10,000 人重新计算结果。
保持检测质量不变，把先验概率降到 0.1%。哪一组人数的变化最关键？
把先验概率保持在 1%，提高特异度。为什么这个控制量影响如此大？
切换到垃圾邮件过滤。哪些概率结构保持不变，哪些决策成本发生变化？

需要检验的误解：检测准确率等于后验概率；阳性就意味着很可能患病；贝叶斯定理与普通计数无关。