Bayes' Theorem: The Equation and Its Friendlier Twin

In the last lesson you solved the medical test by counting people through a tree. That counting was Bayes’ theorem all along — the famous equation is just bookkeeping for exactly the tally you already did. So this lesson holds no new ideas, only new notation: we name the pieces, write the formula, and then — the part that actually changes your life — meet a second form of it that’s so easy you can update in your head while someone is still talking.

First, the four pieces, in the symbols everyone uses. Let $H$ be the hypothesis (“I have the disease”) and $E$ the evidence (“the test was positive”).

• $P(H)$ — the prior: how likely $H$ was before the evidence. (Our base rate, $0.001$.)
• $P(E \mid H)$ — the likelihood: how likely the evidence is if the hypothesis is true. Read the bar “given.” (Our true-positive rate, $0.99$.)
• $P(E \mid \lnot H)$ — the likelihood of the evidence if the hypothesis is false. (Our false-positive rate, $0.01$; $\lnot H$ means “not $H$.”)
• $P(H \mid E)$ — the posterior: how likely the hypothesis is after the evidence. The number we want.

The formula, read off the tree

Here is Bayes’ theorem:

$P(H \mid E) = \frac{P(E \mid H)\,P(H)}{P(E)}$

In words: the posterior equals the likelihood times the prior, divided by how likely the evidence was overall. That denominator $P(E)$ — the total probability of the evidence, from anyone — is just the two ways the evidence can appear, added up:

$P(E) = P(E \mid H)\,P(H) + P(E \mid \lnot H)\,P(\lnot H)$

Now watch this map exactly onto the people you counted last lesson. The numerator $P(E \mid H)\,P(H) = 0.99 \times 0.001 = 0.00099$ is the true-positive branch — 99 people out of 100,000. The denominator’s second term $P(E \mid \lnot H)\,P(\lnot H) = 0.01 \times 0.999 = 0.00999$ is the false-positive branch — 999 people. So:

$P(H \mid E) = \frac{0.00099}{0.00099 + 0.00999} = \frac{0.00099}{0.01098} \approx 0.090$

That’s the same 99 ÷ 1,098 ≈ 9% you got by counting. The fractions are the people, divided by 100,000 top and bottom so the 100,000 cancels. The formula and the tree are the same object. If the symbols ever feel slippery, drop back to counting people — they can never lie to you.

The friendlier twin: the odds form

The formula above is correct but clunky — that denominator is a pain. There’s a second way to write Bayes that throws the denominator away entirely, and it’s the form practitioners actually use. The trick is to stop thinking in probabilities and think in odds (which you met in the last course: odds of 3:1 against rain means 3 “no” for every 1 “yes”).

In odds form, Bayes’ theorem becomes a single multiplication:

$\underbrace{\frac{P(H \mid E)}{P(\lnot H \mid E)}}_{\text{posterior odds}} = \underbrace{\frac{P(H)}{P(\lnot H)}}_{\text{prior odds}} \times \underbrace{\frac{P(E \mid H)}{P(E \mid \lnot H)}}_{\text{likelihood ratio}}$

Say it as a sentence and it’s unforgettable: prior odds × likelihood ratio = posterior odds. You take how the bet stood before, multiply by how strongly the evidence points, and you’ve got how the bet stands now. No denominator, no adding fractions — just one multiply. That middle term, the likelihood ratio (how much more likely the evidence is under $H$ than under $\lnot H$), is the entire strength of the evidence in one number, and it gets a whole lesson next.

Run the medical test through it:

• Prior odds of disease: $0.001 : 0.999$, or about $1 : 1000$ (1 sick for every 1000 healthy).
• Likelihood ratio: $\dfrac{P(\text{positive}\mid\text{sick})}{P(\text{positive}\mid\text{healthy})} = \dfrac{0.99}{0.01} = 99$. A positive is 99 times more likely from a sick person than a healthy one.
• Posterior odds: $1 : 1000$ times $99$ gives $99 : 1000$ — about 99 sick for every 1000 healthy among the positives.

Convert $99 : 1000$ back to a probability: $\dfrac{99}{99 + 1000} = \dfrac{99}{1099} \approx 0.090$, 9% again. Three routes — counting people, the probability formula, the odds form — one answer. The odds form is the one you can do on a napkin.

Worked example: the cab problem, in odds

Let’s run a fresh case end to end to prove the odds form isn’t a one-trick pony. A city’s cabs are 85% Green, 15% Blue. A cab is in a night-time hit-and-run. A witness says it was Blue, and in tests this witness correctly identifies a cab’s colour 80% of the time (and is wrong 20%). How likely is it the cab was actually Blue?

Intuition screams “80% — the witness said so.” Bayes says otherwise. Let $H$ = “the cab was Blue,” $E$ = “the witness said Blue.”

• Prior odds (Blue : Green) $= 15 : 85$, i.e. $3 : 17$.
• Likelihood ratio $= \dfrac{P(\text{says Blue}\mid\text{Blue})}{P(\text{says Blue}\mid\text{Green})} = \dfrac{0.80}{0.20} = 4$. (If the cab is Green, the witness wrongly says Blue 20% of the time.)
• Posterior odds $= 3 : 17$ times $4 = 12 : 17$.
• As a probability $= \dfrac{12}{12 + 17} = \dfrac{12}{29} \approx 0.41$, about 41%.

So even though a witness who’s right 80% of the time said Blue, it’s still more likely the cab was Green — because Green cabs are so common that the witness’s occasional mistakes on them (wrongly saying “Blue”) pile up faster than the genuine Blue sightings. The base rate (prior odds) refuses to be ignored, exactly as it did in the medical test. Same lesson, different costume.

The pieces, in one table

Keep this within reach until the vocabulary is automatic:

When to reach for it

Reach for the odds form whenever you want to update quickly or repeatedly — a string of clues in a negotiation, several symptoms pointing at a diagnosis, multiple independent reviews of a hire. Estimate your prior odds, ask of each new clue “how much more likely is this if I’m right than if I’m wrong?” (that’s the likelihood ratio), and multiply. Reach for the full formula (or the tree) when you want the exact posterior probability for a single piece of evidence, or when you need to show your work. They’re the same theorem; pick the form that fits the job.

Where this goes next

You can now write Bayes three ways and run it in odds on a napkin. One piece is still a black box, though: that likelihood ratio in the middle — the number that says how strong a clue actually is. Next, in “Strength of Evidence,” we open it up: what makes evidence diagnostic versus worthless, why a likelihood ratio near 1 means “ignore this,” how weak clues multiply into strong conclusions, and where the rule extraordinary claims require extraordinary evidence comes from — it’s a likelihood ratio fighting a tiny prior.