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.

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

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