Most people treat beliefs as binary — you believe something or you don't — and change them by flipping. Bayesian reasoning treats beliefs as probabilities and evidence as something that shifts them by an amount proportional to how strong it is. The key insight is that how much a new piece of evidence should move you depends on your prior: how likely the claim was before, given everything else you know.
This explains a notorious trap. Suppose a disease affects 1 in 1,000 people and a test is 95% accurate. You test positive — what's the chance you have it? Most people say around 95%. The real answer is roughly 2%, because the disease is so rare that false positives (5% of the 999 healthy people ≈ 50) vastly outnumber true positives (about 1). The test is good; the base rate dominates. Ignoring base rates is one of the most consequential reasoning errors there is, and it appears everywhere from medical screening to airport security to hiring.
In 1978, researchers at Harvard Medical School posed a version of exactly this problem to 60 physicians, residents, and students at four Harvard teaching hospitals: a disease with a 1-in-1,000 prevalence, a test with a 5% false-positive rate, a patient who tests positive and has no symptoms. What's the probability the patient actually has the disease? Almost half answered 95%. The correct answer is about 2%. Only around a fifth of these highly trained medical professionals got it right. The finding, published in the New England Journal of Medicine, was alarming because these were the people interpreting real test results for real patients. The error is entirely about ignoring the base rate: with 1,000 people, roughly 50 healthy ones get a false positive while about 1 sick person gets a true positive, so a positive result is overwhelmingly likely to be a false alarm.