When intuition surprises you
Start with a familiar puzzle, then learn to ask what information and counting really mean.
- Step 1
Birthday problem
Count pairs rather than comparing everyone with one person.
Checkpoint: Are 23 people being compared with one particular birthday?
No. There are 253 pairs among 23 people. A match anywhere among them is a different event from a match with one specified person.
- Step 2
Events and probability
Make the possible outcomes explicit before counting them.
Checkpoint: Why are two dice sums not equally likely?
Ordered dice pairs are equally likely, but different sums collect different numbers of pairs: one pair gives 2 and six give 7.
- Step 3
Conditioning
Learn how new information changes the sample space.
Checkpoint: Which denominator changes after learning the first die is at least 4?
Keep only the 18 ordered pairs consistent with that information. For a sum at least 10, six qualify, giving 1/3.
- Step 4
Monty Hall
Apply conditioning to a host who knows where the prize is.
Checkpoint: What host behavior makes the usual switching result valid?
The host knows the prize location, always opens an unchosen goat door, and always offers a switch. A different revealing protocol can change the conditional probabilities.
- Step 5
Base rates
Use actual denominators to interpret a positive test.
Checkpoint: Why can a rare condition generate many false positive results?
A small false-positive rate applies to a large unaffected population. Compare true-positive and false-positive counts before interpreting a positive result.
- Step 6
Simpson’s paradox
Discover how group weights can reverse an aggregate comparison.
Checkpoint: Can an aggregate improve without either subgroup improving?
Yes. Shifting population weight toward the higher-rate group changes the weighted average even with unchanged subgroup rates.
After each lesson, try to explain its main result without looking at the formula. Then change one assumption: which part of the answer would change?