The moment one new clue flips the odds entirely.
A car behind one of three doors. You pick; the host opens a goat door. Should you switch?
Recounting inside the world that just shrank — conditional probability.
Experiment
Hands-on experiment
Predict first — you picked a door, and the host (who knows the answer) opened a goat door. Is switching better?
Play it yourself — pick a door
Switch win rate
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Stay win rate
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Read more — why it exists · insights · common mistakes · formulasExpand ▾
Why
Why does this exist?
When this problem ran in a magazine in 1990, a thousand PhDs mailed in protests against the correct answer ('switch'). A monument to how helpless human intuition is before conditional probability.
The key is information. The moment the host opens a door you've learned something, and the list of possible worlds has changed. Recounting in the changed list — that's all conditional probability is.
Interpreting medical tests, weighing courtroom evidence, filtering spam — wherever 'probability after learning something' matters, this thinking decides the judgment.
Insight
Insights from the video
“A condition doesn't change the probability — it changes the world you count in.”
P(A|B) is 'A, recounted inside the world where B happened'. What changes is the denominator, not the numerator — the heart of conditional probability.
“Monty Hall's real key: the host KNOWS.”
If the host didn't know and opened a goat door by luck, the odds genuinely become 50:50. The same scene with a different information source has a different probability — why conditional probability is delicate.
Misconception
Common misconceptions
Two doors remain, so it's 50:50.
Your first pick's 1/3 chance doesn't change when the host opens a door. The remaining 2/3 piles onto the other door — because the host knows the answer and only ever opens goat doors.
A probability, once set, never changes.
Probability is a value about what you know. New information shrinks the set of possible worlds, and recounting in the smaller set changes the number. That is conditional probability.
Formula
Writing it as math
Write the 'world-shrinking' you experienced as a formula and you get the definition.
Definition of conditional probability
The probability of A given that B happened. The denominator shrinks from everything to B — recounting in the smaller world.
The Monty Hall computation
The switching strategy wins exactly when the first pick was wrong. That happens with probability 2/3, so switching wins 2/3 of the time.
What independence means
If learning the condition doesn't move the number, the events are independent. This is why a coin's past can't change its future (the Probability lab).
In Real Life
Where you meet it in real life
Reading medical tests
'The probability of disease given a positive test' is conditional. That it can differ wildly from intuition leads straight to Bayes' theorem.
Spam filters
'The probability of spam given this word' — the skeleton of email filtering is conditional probability.
Courtroom evidence
Confusing 'P(guilty | evidence)' with 'P(evidence | guilty)' is the famous prosecutor's fallacy.
Games and cards
In poker, every bet (information) from your opponent changes the denominator of your win-rate calculation.
Try Yourself
Test yourself
Q1You roll a die and hear the hint 'it's 3 or higher'. What's the probability it's a 6?Show answer ▾
1/4. The world shrank to {3,4,5,6} — four outcomes, one of which is 6. Before the hint 1/6, after 1/4 — the denominator changed.
Q2A family has two children and you learn 'at least one is a girl'. What's the probability both are girls? (It's not 1/2.)Show answer ▾
1/3. The possible worlds are (G,G)(G,B)(B,G) — three, one of which works. The key is recounting in the world where only (B,B) was removed.
Q3Monty Hall with 100 doors: you pick one, the host opens 98 goat doors. Switch?Show answer ▾
Absolutely — switching wins 99/100. Amusingly, the more doors there are, the closer intuition gets to the right answer ('switch'). Three doors is the same structure.
Try answering yourself before revealing it — getting it wrong is where learning starts.
Connection
Concepts connect
Previous concept
Probability
With the basics of counting uncertainty in hand, it's time to see how the numbers move when information arrives.
← Probability labLeads to next
Bayes' Theorem
Flip a conditional probability (from P(A|B) to P(B|A)) and you get Bayes — where the shock of 'positive test, yet only 9% chance of disease' awaits.
Go to the Bayes' Theorem lab →Related
Labs worth exploring together
Related lab
Probability
The world before conditions — build the base intuition (law of large numbers, independence) first.
Go to the Probability lab →