Probability After New Information

Restrict the world you are counting

A probability is a number from 0 to 1 describing uncertainty. A conditional probability, written conceptually as P(A given B), asks how likely A is after learning B. The word given restricts the denominator to cases where B is true.

1,000 deployments
100 used a risky configuration
 30 of those risky deployments failed
900 did not use it
 45 of those other deployments failed

P(failure | risky) = 30 / 100 = 30%
P(failure | not risky) = 45 / 900 = 5%

Do not confuse P(failure given risky) with P(risky given failure). Of the 75 total failures, 30 were risky, so the reverse probability is 30/75 = 40%. Same table, different restricted group, different question.

Analogy: Asking what fraction of red cards are kings? first selects red cards. Asking what fraction of kings are red? first selects kings. Reversing the condition reverses the pile you count.

Warning: Natural-language claims often reverse these probabilities. State the conditioned group explicitly before calculating.