Imagine you take a covid test that’s 99% accurate and it comes back positive. The intuitive answer is “I almost certainly have covid.” The actual answer, when only 1 in 100 people in your area has it, is closer to 1 in 6.
Bayes’ rule is the formula that makes this make sense — it’s how you update a belief when new evidence arrives, and it’s the trick every quant interviewer reaches for first. We’ll work through the covid case in the next four steps, building the formula one piece at a time. By the end you’ll see why “99% accurate” means almost nothing without the prior — and why missing the prior is the single most common interview wrong-answer.
✓ Intro · expand
Try first (productive failure)
Before the worked example: spend 60 seconds taking your best shot at this.
A guess is fine — being briefly wrong about a problem makes the explanation
land harder when you read it. This appears once per tutorial; skip
if you already know the trick.
60s
✓ Try first · expand
Worked example
A disease affects 1% of the population. A test is 99% sensitive ($P(+ \mid D) = 0.99$) and 95% specific ($P(- \mid D^c) = 0.95$, so $P(+ \mid D^c) = 0.05$). A randomly chosen person tests positive. What is the probability they actually have the disease?
✓ Worked example · expand
Practice 1 of 3Type a fraction, decimal, or expression — mathjs parses it.
✓ Practice · expand
Reflection
In your own words, what’s the cue in a problem statement that tells you to reach for Bayes rather than just the multiplication rule? And why does a “99% accurate” test on a rare disease produce a posterior near 1/6 instead of 99% — what work is the prior doing in the denominator?