Problems

278 problems

Probability, statistics and reasoning problems of the kind asked in quant researcher interviews. Free, no account. Each problem has hints and a worked solution.

By topic: probability · random processes · order statistics · inference · regression · linear algebra · games · puzzles

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1 The blue-eyed islanderspuzzles puzzles medium
2 When adjusted R² stops fallingregression regression medium
3 Ants on a metre stickpuzzles puzzles easy
4 A bee between two trainspuzzles puzzles easy
5 Ten discs on three pegspuzzles puzzles easy
6 An unbiased chance of a quiet minuteinference inference hard
7 Taking turns until head then tailrandom processes random processes medium
8 How many draws to pass 0.95order statistics order statistics easy
9 Cutting a cube into 27puzzles puzzles easy
10 A die that decides how many diceprobability probability easy
11 Two signals, one R²regression regression medium
12 Splitting coins in the darkpuzzles puzzles easy
13 The area under a Brownian pathrandom processes random processes medium
14 R² from pure noiseregression regression medium
15 The snail and the wallpuzzles puzzles easy
16 Always ahead in the countrandom processes random processes medium
17 First to roll a sixprobability probability easy
18 Every row twiceregression regression medium
19 Winning at crapsprobability probability medium
20 A clock gone madpuzzles puzzles medium
21 Every face at least onceprobability probability easy
22 One hundred hats in a linepuzzles puzzles medium
23 Four coins and one second chanceprobability probability easy
24 A twelve or two sevensrandom processes random processes medium
25 A weighted geometric seriespuzzles puzzles easy
26 Guessing a quizprobability probability easy
27 How many stocks are enoughlinear algebra linear algebra easy
28 One free restartgames games hard
29 A regressor measured with noiseregression regression easy
30 Back at the start of a tetrahedronrandom processes random processes medium
31 Five p-values and a false discovery rateinference inference medium
32 The least risky mixlinear algebra linear algebra easy
33 Chuck-a-luckgames games easy
34 Walking down the escalatorpuzzles puzzles medium
35 Three hundred parachutesinference inference medium
36 The larger of two normalsorder statistics order statistics medium
37 Bricks in a boxpuzzles puzzles hard
38 One hundred in a circlepuzzles puzzles medium
39 Calling the colour of each cardgames games medium
40 How often the hands meetpuzzles puzzles easy
41 Information in sixteen waiting timesinference inference medium
42 The first total past 100random processes random processes hard
43 Reaching one before time onerandom processes random processes medium
44 A random quadraticprobability probability easy
45 An even productprobability probability easy
46 Two looks at one pathrandom processes random processes easy
47 Digits that strictly increasepuzzles puzzles easy
48 Four tyres and a sparepuzzles puzzles easy
49 Positive at both timesrandom processes random processes medium
50 Dividing by n − 1, n or n + 1inference inference medium
51 Correlation after cutting the sampleinference inference hard
52 Guess my number for moneygames games medium
53 Hiring the best of fivegames games medium
54 Soft thresholdingregression regression medium
55 Both sixes, given oneprobability probability easy
56 Chords between six pointspuzzles puzzles medium
57 The last digit of 17 to the 17thpuzzles puzzles easy
58 Mean or maximum for the endpointinference inference medium
59 Mean absolute value of a normalinference inference easy
60 Paths across a gridpuzzles puzzles easy
61 Paying to open boxesgames games easy
62 One heavy coin in twenty-sevenpuzzles puzzles easy
63 An ant across a cubepuzzles puzzles easy
64 Who keeps the ballrandom processes random processes medium
65 Nobody gets their own hatprobability probability easy
66 Three doorsprobability probability easy
67 A coin rolling around a coinpuzzles puzzles easy
68 Twenty tests, one false alarminference inference easy
69 Visiting every hourrandom processes random processes hard
70 A regression line in a cornerregression regression medium
71 The longest pieceorder statistics order statistics medium
72 Updating an event rateinference inference easy
73 How many recordsorder statistics order statistics easy
74 Three cards in a rowgames games medium
75 Two random directions in spacelinear algebra linear algebra medium
76 Waiting for HTHTrandom processes random processes medium
77 A book you cannot losegames games easy
78 The card after the first aceprobability probability medium
79 Who gets the first aceprobability probability medium
80 Newton and Pepysprobability probability medium
81 Averaging Cauchy datainference inference medium
82 The Gaussian integralpuzzles puzzles medium
83 Meeting for lunchprobability probability easy
84 Handshakes without crossingpuzzles puzzles medium
85 Five sailors and the coconutspuzzles puzzles hard
86 Neighbours of the same rankprobability probability easy
87 Identity plus an outer productlinear algebra linear algebra easy
88 Distances along a linelinear algebra linear algebra hard
89 The option to buy after inspectiongames games easy
90 Three ants on a triangleprobability probability easy
91 A flat line through a curveregression regression medium
92 One step from the edgerandom processes random processes medium
93 Two sixes in a rowrandom processes random processes medium
94 Repainting until all matchrandom processes random processes medium
95 A day without callsinference inference medium
96 One odd pair among fourlinear algebra linear algebra hard
97 A uniform inside a uniformprobability probability medium
98 Double six for a hundredgames games easy
99 St Petersburg with a finite bankgames games medium
100 Accept or pass the diegames games medium
101 A white face on a painted cubeprobability probability medium
102 A d30 against a d20 with a rerollgames games easy
103 Three points, one halfprobability probability medium
104 Arranging balls in two boxesprobability probability medium
105 Ones against fivesinference inference medium
106 Tying noodles into loopsprobability probability hard
107 Where the smallest fallsorder statistics order statistics easy
108 Five cards of one suitprobability probability easy
109 Heads times tailsinference inference medium
110 The last hour on the clockrandom processes random processes medium
111 Twos and minus oneslinear algebra linear algebra easy
112 Four points around the centreprobability probability hard
113 Poker with three cardsgames games hard
114 A knight finds its way homerandom processes random processes hard
115 A hole through a ballpuzzles puzzles medium
116 How many coins in three weighingspuzzles puzzles hard
117 Variance of a two-regime returninference inference easy
118 Four people, one torchpuzzles puzzles medium
119 A Chebyshev bound for a spreadinference inference easy
120 Two thermometersinference inference easy
121 Waiting for the first red ballprobability probability medium
122 Chocolates and jelly beansrandom processes random processes easy
123 Paying for the firewoodpuzzles puzzles easy
124 Two numbers with no common factorprobability probability hard
125 Lily pads with a head startpuzzles puzzles easy
126 Left out of the bootstrapinference inference easy
127 Adding until past oneprobability probability medium
128 Rock-paper-scissors without rockgames games medium
129 Coming back before the endrandom processes random processes medium
130 Positive eights, negative fivespuzzles puzzles hard
131 How many different birthdaysprobability probability medium
132 Variance of an Itô integralrandom processes random processes medium
133 Digit sums of a huge powerpuzzles puzzles hard
134 Twenty-five horses, five lanespuzzles puzzles medium
135 Two crossing pillarspuzzles puzzles hard
136 Cycles of a shuffleprobability probability hard
137 A boy born on a Tuesdayprobability probability medium
138 Shift, invert, cubelinear algebra linear algebra easy
139 Seven pays, even costsgames games easy
140 Breaking a stick into a triangleorder statistics order statistics easy
141 Down two with one shot leftgames games easy
142 Three daughters and a house numberpuzzles puzzles medium
143 Back at twelve after twelve stepsrandom processes random processes medium
144 Two-finger Morragames games medium
145 Seven heads and a flat priorinference inference easy
146 Three signals that overlapregression regression medium
147 Up to three rollsgames games medium
148 A series that goes the distanceprobability probability easy
149 Two eggs, one hundred floorspuzzles puzzles medium
150 A car on a quiet roadprobability probability easy
151 A clock with twin handspuzzles puzzles hard
152 Calls that become salesprobability probability medium
153 Five pirates and a hundred coinspuzzles puzzles medium
154 Average speed of a round trippuzzles puzzles easy
155 Red pays, black costsgames games hard
156 Sixty heads in a hundredinference inference easy
157 Smaller over largerorder statistics order statistics medium
158 Rectangles on a chessboardpuzzles puzzles easy
159 The two best players in the finalprobability probability medium
160 A pair or better in five cardsprobability probability easy
161 Drying potatoespuzzles puzzles easy
162 Six straight cutspuzzles puzzles easy
163 Split and discard the smaller pilegames games medium
164 How far from the North Poleprobability probability medium
165 The sample standard deviation is biasedinference inference hard
166 The largest 0-1 determinantlinear algebra linear algebra medium
167 A walk on the keypadrandom processes random processes easy
168 How low can the third correlation golinear algebra linear algebra medium
169 The spread of five uniformsorder statistics order statistics medium
170 Two regression slopesregression regression medium
171 How much to betgames games easy
172 Beta is not correlationregression regression easy
173 A needle on a ruled floorprobability probability medium
174 Bidding on what the seller knowsgames games easy
175 Four weights for forty poundspuzzles puzzles medium
176 Two points on a circleprobability probability medium
177 Sealed bids, three biddersgames games medium
178 Building three correlated normalslinear algebra linear algebra medium
179 The average squared determinantlinear algebra linear algebra medium
180 Counting tanks from serial numbersinference inference medium
181 The mean of exp of Brownian motionrandom processes random processes easy
182 The red candies run out firstprobability probability medium
183 One extra coinprobability probability medium
184 Twos on the diagonallinear algebra linear algebra medium
185 The sum up to the first sixrandom processes random processes easy
186 x years old in the year x squaredpuzzles puzzles easy
187 Three different facesprobability probability easy
188 How negative can it belinear algebra linear algebra medium
189 A wand that turns two coins overgames games easy
190 Is strategy B betterinference inference easy
191 Positive correlation, negative coefficientregression regression medium
192 The option to double the stakegames games medium
193 The Chevalier's double sixesprobability probability easy
194 An acute triangle from a broken stickorder statistics order statistics hard
195 Two rolls two apartrandom processes random processes medium
196 Across the cuberandom processes random processes medium
197 A hub and nine spokeslinear algebra linear algebra medium
198 Joining five chainspuzzles puzzles medium
199 Two ferriespuzzles puzzles medium
200 A hundred-sided die, one dollar per rerollgames games medium
201 Smaller over larger, for normalsorder statistics order statistics hard
202 Betting on the series one game at a timegames games medium
203 Real eigenvalues by chancelinear algebra linear algebra hard
204 Lost in a caverandom processes random processes easy
205 i to the power ipuzzles puzzles easy
206 The last of three serversorder statistics order statistics easy
207 How far apart two dice landprobability probability easy
208 The sixth moment of Brownian motionrandom processes random processes easy
209 Exactly half headsprobability probability medium
210 A ratio between one and twoprobability probability easy
211 As many ones as possible, square zerolinear algebra linear algebra hard
212 Change for a dollarpuzzles puzzles medium
213 The larger of two diceprobability probability easy
214 The last passenger's seatprobability probability medium
215 The angle at a quarter past threepuzzles puzzles easy
216 One poisoned bottle in a thousandpuzzles puzzles easy
217 Keep rolling while it is highgames games easy
218 A positive testprobability probability easy
219 A rope around the Earthpuzzles puzzles easy
220 Estimating the endpointinference inference easy
221 Mean of the upper tailinference inference medium
222 Four doors, two chances to switchgames games medium
223 Fish in the lakeinference inference easy
224 Two sticks, two measurementsinference inference medium
225 A sum above one, a product below 3/16probability probability hard
226 The camel and the bananaspuzzles puzzles hard
227 Three boxes of coinsprobability probability easy
228 Four gallons from a 3 and a 5puzzles puzzles medium
229 Which coin was it?probability probability easy
230 Three sticks, one triangleorder statistics order statistics medium
231 Only even rolls before the sixprobability probability hard
232 Five values and the medianorder statistics order statistics easy
233 Breaking a chocolate barpuzzles puzzles easy
234 Redraw or keepgames games hard
235 Correlation of the smaller and the largerorder statistics order statistics medium
236 Against the driftrandom processes random processes hard
237 Colour runs in a shuffled deckprobability probability easy
238 Win by tworandom processes random processes easy
239 Swapping the factorslinear algebra linear algebra medium
240 Where the sample correlation is most biasedinference inference medium
241 Will the amoebas die out?random processes random processes medium
242 How many onespuzzles puzzles easy
243 Nim with 3, 4 and 5games games easy
244 Closer to the centre than to the edgeprobability probability hard
245 Guess the total, win the totalgames games easy
246 Two tetrahedral dice until a tiegames games easy
247 Waiting for tail then headrandom processes random processes easy
248 One hundred boxespuzzles puzzles hard
249 Four strange dicegames games medium
250 The postman who likes full boxesprobability probability hard
251 The two best of three dicegames games medium
252 Furthest along a directionlinear algebra linear algebra easy
253 Closest to a loaded diegames games medium
254 Calling the next red cardrandom processes random processes medium
255 Counting head pairsprobability probability easy
256 An infinite tower of powerspuzzles puzzles medium
257 A three-way duelgames games hard
258 The missing regressorregression regression medium
259 How long the walk lastsrandom processes random processes easy
260 Spin again or notprobability probability easy
261 Seated in order of ageprobability probability easy
262 The size of a coin testinference inference easy
263 A race between two patternsrandom processes random processes hard
264 The gap you walk intoprobability probability medium
265 Red balls pay, the yellow one ruinsgames games medium
266 One hundred lockerspuzzles puzzles easy
267 A random symmetric matrixlinear algebra linear algebra medium
268 A shared birth monthprobability probability easy
269 Confidence that feeds on itselfrandom processes random processes medium
270 All the digits up to a millionpuzzles puzzles medium
271 Zeros at the end of 100!puzzles puzzles easy
272 Paint on a big cubepuzzles puzzles easy
273 Doubles wipe you outgames games medium
274 Ruin with an edgerandom processes random processes medium
275 How long to trust a Sharpe of 1inference inference easy
276 Digits adding to nineteenpuzzles puzzles hard
277 How many to askinference inference easy
278 Two R² values, one joint fitregression regression medium

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