Concept
Conditional Probability
The probability of something after new information changes what is still possible. The important move is to update the denominator: cases ruled out by the new information should no longer be counted. Many mistakes happen when people keep using the original odds after the situation has narrowed.
Lessons
New Information Changes the Odds
Conditional probability is about updating the sample space after you learn something.
Rare Things Stay Rare
Even strong evidence can mislead when the thing you are testing for is very uncommon.
Small P Is Not Proof
A small p-value says the data would be surprising under a null model, not that the claim is probably true or important.
Related Facts
Risk & Decisions
4 min · medium
A one-in-a-million match is not a one-in-a-million chance of innocence
When a forensic trait matches 1 in a million people, a big enough population still holds many innocent matches — the match probability is not the probability of innocence.
Risk & Decisions
4 min · medium
A good test can still scare too many people
When a condition is rare, most positive test results can still be false positives.
Data Tricks
4 min · medium
A p-value is not the chance your result is wrong
A p-value is a check for surprise: it asks how often a result this extreme would appear if there were really no effect. It does not tell you the chance your result is wrong.
Paradoxes
4 min · easy
The door you did not pick is probably better
In Monty Hall, switching doors wins about twice as often as staying.