Level 3: Expectations
Linearity, the tail-sum formula, first-step analysis, and waiting times — the workhorses behind every expected-value desk question.
36 problems
First 5 free · rest Premium
- 1→
Expected Maximum of Two Dice
●●○○○Easy~4 min
- 2→
Expected Wait for the First Head
●●○○○Easy~4 min
- 3→
The Coupon Collector
●●●○○Medium~6 min
- 4→
The Hat-Check Problem
●●●○○Medium~5 min
- 5→
HH Before vs. HT
●●●●○Hard~7 min
- 6Premium
Expected Minimum of Uniforms
●●●○○Medium~5 min
- 7Premium
How Many Uniforms to Exceed 1?
●●●●○Hard~7 min
- 8Premium
Counting Record Highs
●●●○○Medium~6 min
- 9Premium
Distinct Faces in n Rolls
●●●○○Medium~5 min
- 10Premium
Waiting for Two Sixes
●●○○○Easy~4 min
- 11Premium
Expected Gap Between Two Dice
●●○○○Easy~4 min
- 12Premium
The Larger Half of a Broken Stick
●●○○○Easy~4 min
- 13Premium
Where Does the k-th Smallest Land?
●●●○○Medium~6 min
- 14Premium
Waiting for the Pattern HTH
●●●●○Hard~8 min
- 15Premium
Counting Inversions
●●●○○Medium~5 min
- 16Premium
First of Several Geometric Clocks
●●○○○Easy~4 min
- 17Premium
The St. Petersburg Game
●●●●○Hard~6 min
- 18Premium
Expected Empty Boxes
●●●○○Medium~5 min
- 19Premium
Expected Product of Two Dice
●○○○○Warm-up~2 min
- 20Premium
Cycles in a Random Permutation
●●●○○Medium~6 min
- 21Premium
How Many Until a Birthday Match?
●●●●○Hard~7 min
- 22Premium
A Random Number of Dice
●●○○○Easy~4 min
- 23Premium
Average Distance on a Segment
●●○○○Easy~4 min
- 24Premium
All Coins Heads at Once
●●○○○Easy~4 min
- 25Premium
Expected Larger of Two Uniforms
●●○○○Easy~3 min
- 26Premium
Expected Number of Peaks
●●●●○Hard~6 min
- 27Premium
Waiting for Double Sixes
●●○○○Easy~3 min
- 28Premium
The Inspection Paradox
●●●●○Hard~7 min
- 29Premium
Triangles in a Random Graph
●●●●○Hard~5 min
- 30Premium
Average of a Reciprocal
●●●○○Medium~4 min
- 31Premium
Expected Comparisons in Quicksort
●●●○○Medium~6 min
- 32Premium
Expected Distinct Birthdays
●●●○○Medium~5 min
- 33Premium
Expected Ascents in a Shuffle
●●●○○Medium~4 min
- 34Premium
Shuffling Until Sorted
●●○○○Easy~3 min
- 35Premium
Coupon Collector with Unequal Odds
●●●●○Hard~7 min
- 36Premium
Variance of Matches
●●●●○Hard~6 min