Probability
Counting combinations
Count choices, codes, groups and routes by checking order and repeats.
6 questions · 3 levels · No time limit
Probability
Count possible outcomes and reason about chance.
Try a practice
Probability
Reason with probability
Compare outcomes, follow dependent draws and find expected point values.
8 questions · 3 levels · Untimed
Probability
Conditional probability
Work through restricted sample spaces, linked selections and conditional events. Explore the counts behind each answer.
1 question · 3 levels · Untimed
Probability
Switch or stay
Open the doors, inspect the host’s rule and test when switching really helps.
3 steps · 3 levels · Untimed
Probability
Risk and reward
Try point games and discover when a big prize, a better average and a better chance mean different things.
3 steps · 3 levels · Untimed
Probability
Birthday collisions
How small can a group be before matching birthdays become likely? Predict, generate groups and inspect the matches.
3 steps · 3 levels · Untimed
Probability
Pattern races
Race coin patterns and discover why equally likely sequences can have different waiting times.
3 steps · 3 levels · Untimed
Probability
Nontransitive dice
Find dice that beat one another in a cycle, then test how rolling twice changes the best response.
3 steps · 3 levels · Untimed
Learn a method
Browse lessonsLesson
Count equally likely outcomes
Keep outcomes equally likely before dividing favorable cases by all cases.
3 min · Example and practice
Lesson
Decide whether order matters
Separate an ordered arrangement from a group that contains the same items.
3 min · Example and practice
Lesson
Condition on information
Use natural counts to choose the right conditional denominator.
5 min · Example and practice
Lesson
Update after a draw
Track the changing counts when an object is not replaced.
5 min · Example and practice
Lesson
Follow the host rule
Explain why the same empty door can lead to different odds.
6 min · Example and practice
Lesson
Compare expected rewards
Subtract the cost and weight every net outcome by its probability.
5 min · Example and practice
Lesson
Count winning sequences
Use complements and disjoint cases to compare multi-roll targets.
6 min · Example and practice
Lesson
Count birthday collisions
Use the chance of no match to understand why shared birthdays arrive surprisingly soon.
5 min · Example and practice
Lesson
Why patterns overlap
Separate a pattern’s chance in one window from the time you wait for it.
6 min · Example and practice
Lesson
Race patterns with a biased coin
Track both patterns on one sequence and count unfinished races.
7 min · Example and practice
Lesson
Compare dice in pairs
Discover why a die can beat one rival and lose to another.
6 min · Example and practice
Lesson
Choose a die after two rolls
Build the distribution of sums before choosing a counter to an opponent.
7 min · Example and practice
Browser storage is unavailable. Changes last only on this page.