Reasoning with Probability

Introduction

People often misjudge probabilities that involve evidence. A famous example: a medical test is 90 percent accurate and you test positive. Are you 90 percent likely to have the condition? Usually not. This lesson explains why, using numbers you can check. It is not medical advice, and real tests and conditions vary.

Key terms

• Prevalence or base rate: the share of people who have the condition. Say 1 percent.
• Sensitivity: the probability that the test is positive if you have the condition. Say 90 percent.
• Specificity: the probability that the test is negative if you do not have it. Say 95 percent, so the false positive rate is 5 percent.

The calculation with natural frequencies

Imagine 10,000 people.

• With a 1 percent base rate, 100 have the condition, and 9,900 do not.
• Of the 100 who have it, 90 percent test positive: 90 true positives.
• Of the 9,900 who do not, 5 percent test positive: 495 false positives.
• Total positives: 90 + 495 = 585.

If you test positive, the chance you have the condition is 90 / 585 ≈ 15.4 percent.

Most positives are false because the condition is rare and the 5 percent false-positive rate applies to a large healthy group. This is base-rate neglect: ignoring how common the condition is.

Bayes' rule

The same result in formula form:

P(condition | positive) =
  P(positive | condition) x P(condition)
  ---------------------------------------------------------
  P(positive | condition) x P(condition) + P(positive | no condition) x P(no condition)

Numbers: (0.9 x 0.01) / (0.9 x 0.01 + 0.05 x 0.99) = 0.009 / (0.009 + 0.0495) = 0.009 / 0.0585 = 0.1538.

This rule shows how to update a belief (the prior) with evidence (the test) to a new belief (the posterior).

What changes the answer

• A higher base rate (for example a high-risk group where it is 20 percent) gives a much higher chance after a positive result.
• A better specificity greatly reduces false positives.
• Repeat or confirmatory tests raise confidence, if the tests are independent.

Doctors use this reasoning when deciding who to screen and how to interpret results.

Other places this matters

• Spam filters, fraud alerts and security alarms: most alerts may be false because real events are rare.
• Legal reasoning: the "prosecutor's fallacy" confuses P(evidence | innocent) with P(innocent | evidence).
• Forecasting: updating predictions as news arrives.
• Everyday claims: "this rare symptom suggests this rare disease" needs both base rates.

Good habits for uncertain evidence

• Ask for the base rate.
• Translate to counts out of 1,000 or 10,000 people.
• Distinguish P(A given B) from P(B given A). They are not the same.
• Update gradually and stay open to changing your mind.
• Remember models are simplified; real tests and populations vary.

Try it yourself

Recompute the example with a base rate of 10 percent and then 0.1 percent, keeping the same test. Use 10,000 people each time. Write one sentence on what you learn.

Summary

• Sensitivity, specificity and base rate together determine what a positive result means.
• With a rare condition, most positives can be false, as natural frequencies show.
• Bayes' rule updates a prior belief with evidence, and base rates must never be ignored.