Classical and conditional probability

Idea

Conditional probability is the probability that $A$ happens, given that $B$ has happened. The sample space shrinks to the inside of $B$.

All three modes ask the same question: what is the sample space right now. Conditional probability cuts it down to $B$. In Monty Hall, the host’s choice makes it uneven. The central limit theorem changes what is being observed, from a single trial to the average of $n$ trials. Intuition usually goes wrong because it is still using the old sample space to answer a new question.

Interaction

• Conditional probability: drag the proportions of events $A$, $B$, and their intersection, and watch how $P(A\mid B)$ changes once the sample space is cut down to $B$
• Central limit theorem: set the sample size and speed and build up the statistics. Watch the average of repeated trials settle into a bell shape
• Monty Hall: try switching and staying again and again, and use the running statistics to compare the long-run win rates with intuition

Suggested order: conditional probability → central limit theorem → Monty Hall.

What to notice

• All three modes change the sample space: cut it down to $B$, let new information make it uneven, or look at the average of $n$ trials instead of a single trial.
• Intuition usually goes wrong by answering a new question with the old space. Once the proportions in the figure change, the old “uniform” reading no longer holds.
• A single trial does not have to look like a bell. Once the thing being observed becomes the average, the same probability mass piles up again.

Related

• Conditional probability and Bayes’ theorem
• Binomial distribution to the normal distribution
• Buffon’s needle

Further reading

• Bayes’ theorem (Wikipedia)
• Central limit theorem (Wikipedia)
• Monty Hall problem (Wikipedia)