Conditioning as a Problem-Solving Tool

First-step analysis

Monty Hall

Simpson's paradox and hidden partitions

What feature of the standard Monty Hall setup makes switching win with probability 2/3?

In the hospital example, hospital A wins overall while B wins within each difficulty slice. What exactly produced the overall reversal?

A symmetric random walk starts at 0. Condition on the first step: it moves to +1 or to -1, each with probability one half. What is the probability it moves to +1? Enter a decimal like 0.5.

Monty opened door 3. The joint weights for prize behind door 1 and door 2 are 1/6 and 1/3. What is the probability the prize is behind door 2? Write it as a fraction and enter the numerator.

Hospital A: 90 of 100 easy cases succeed. Hospital B: 10 of 10 easy cases succeed. Among easy cases only, which raw fraction is higher, and by how much in percentage points? Enter the difference as a number like 10.

In your own words, state the most important definition, theorem or method of this lesson and explain what it says.