Unit 4: Probability, Random Variables & Probability Distributions

Conditional Probability

Applying probability rules, including conditional probability, to compound events.

Advanced25 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Why a Positive Test Result Isn't as Certain as It Feels

A medical test correctly flags 45 of the 50 people who truly have a rare disease, and incorrectly flags only 19 of the 950 healthy people, out of 1000 people tested. Before reading on: given a positive test result, what's your gut estimate for the chance that person actually has the disease?

Definition — Conditional Probability

P(A|B) = P(A and B)/P(B) — the probability of A, given that B has already happened, restricts attention to only the outcomes where B occurred. Two events are independent exactly when P(A|B) = P(A) — knowing B happened doesn't change A's probability at all.

P(A \mid B) = \frac{P(A \text{ and } B)}{P(B)}

	Test Positive	Test Negative	Total
Has Disease	45	5	50
No Disease	19	931	950
Total	64	936	1000

1000 people tested for a rare disease.

Worked Example — Computing a Conditional Probability from a Two-Way Table

P(has disease | positive test) = P(disease and positive)/P(positive) = (45/1000)/(64/1000) = 45/64 ≈ 0.703, or about 70.3%. Even with a test that correctly flags 90% of actual cases (45/50), a positive result only means about a 70% chance of truly having the disease — because the disease is rare (5% of the population), most positive results still come from the much larger group of healthy people, a small fraction of whom are false positives.

Worked Example — Checking Independence

Is testing positive independent of having the disease? P(positive) = 64/1000 = 0.064. P(positive | disease) = 45/50 = 0.9. Since 0.9 ≠ 0.064, the events are strongly dependent — exactly what a useful medical test should show: knowing someone has the disease dramatically changes the probability their test is positive.

Probability Simulator

Experiment

Type

Sides

Number of dice

Run a trial to see the outcome here.

Simulation

Target trials

Trials per tick

Track outcome

0 / 200 trials

Histogram — experimental vs theoretical

Running probability — Law of Large Numbers

Statistics

Mean

— experimental

3.5 theoretical

Variance

— experimental

2.917 theoretical

Standard deviation

— experimental

1.708 theoretical

Trials

0 experimental

— theoretical

Frequency table

Outcome	Experimental	Theoretical
1	—	0.167
2	—	0.167
3	—	0.167
4	—	0.167
5	—	0.167
6	—	0.167

Central Limit Theorem

Draws 300 independent samples of 5trials each from the current experiment, averages each sample, and histograms the resulting sample means — the distribution should look progressively more bell-shaped (normal) as the sample size grows, even though the experiment’s own distribution usually isn’t.

Sample size

Number of samples

Tip

When a conditional probability feels surprising, check whether the event being conditioned on (here, having the rare disease) is itself rare — a low base rate is almost always the reason a seemingly accurate test still produces many false positives relative to true positives.

Common Mistakes

Confusing P(positive | disease) — the test's accuracy on people who have the disease — with P(disease | positive) — the actual chance a positive result reflects the disease.
These are different conditional probabilities, computed in opposite directions, and the worked example shows they can differ substantially (90% vs. 70.3%) — always check which event is being conditioned on which.

Key Takeaways

P(A|B) restricts attention to only the outcomes where B occurred, then asks what fraction of those also satisfy A.
P(A|B) and P(B|A) are generally different quantities — order matters in conditional probability.
Independence means P(A|B) = P(A): learning B occurred doesn't change A's probability at all.

Summary

Conditional probability reasons about specific events. The next lesson builds a fuller mathematical object — a random variable — that assigns probabilities to every possible numerical outcome at once.

Track your progress

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