Conditional probability & Bayes' rule

Each square is one of 100 people. Teal means they have the condition and tested positive; ochre means they don't have it but tested positive anyway. Drag the base rate down — watch the ochre squares start to outnumber the teal ones, even though the test itself hasn't changed at all.

$P(A\mid B)$ reads "the probability of $A$, given that $B$ happened" — your belief in $A$ updated by evidence $B$. Bayes' rule says how to compute it —

• $A$ — the event we want to know about (e.g., "has the condition").
• $B$ — the observed evidence (e.g., "tested positive").
• $P(A\mid B)$ — the updated probability of $A$ given that $B$ happened (the posterior).
• $P(B\mid A)$ — the probability of seeing evidence $B$ if $A$ is true (the likelihood, e.g. sensitivity).
• $P(A)$ — the probability of $A$ before seeing any evidence (the prior, or base rate).
• $P(B)$ — the overall probability of observing $B$, across all cases.

1. Map A and B to the demo

   Here $A$ is "has the condition," $B$ is "tested positive."

2. Read the likelihood and the prior

   $P(B\mid A)$ is the test's sensitivity — how often it catches a real case. $P(A)$ is the base rate you just dragged.

Watch $P(A\mid B)$ track the base rate: the test's accuracy never changes, yet how much a positive result should actually convince you depends heavily on how rare the condition was to begin with.

Base rate $1\%$, sensitivity $90\%$, false-positive rate $10\%$:

1. Find the overall chance of a positive test

   $$P(B) = 0.9(0.01) + 0.1(0.99) = 0.009 + 0.099 = 0.108$$
2. Apply Bayes' rule

   $$P(A\mid B) = \frac{0.9 \times 0.01}{0.108} \approx 8.3\%$$
3. Read what it means

   A "90% accurate" test, and a positive result still means less than a 1-in-10 chance of actually having it — because the condition was rare enough that false alarms from the other 99% swamp the true positives.