Probability interview questions

60 problems

Dice, coins, cards and urns: conditional probability, expectation, independence and counting. These are the questions that open most quant researcher interviews, and the ones where a clean setup matters more than heavy computation. Each problem has hints and a worked solution, free, no account.

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10 A die that decides how many dice easy
17 First to roll a six easy
19 Winning at craps medium
21 Every face at least once easy
23 Four coins and one second chance easy
26 Guessing a quiz easy
44 A random quadratic easy
45 An even product easy
55 Both sixes, given one easy
65 Nobody gets their own hat easy
66 Three doors easy
78 The card after the first ace medium
79 Who gets the first ace medium
80 Newton and Pepys medium
83 Meeting for lunch easy
86 Neighbours of the same rank easy
90 Three ants on a triangle easy
97 A uniform inside a uniform medium
101 A white face on a painted cube medium
103 Three points, one half medium
104 Arranging balls in two boxes medium
106 Tying noodles into loops hard
108 Five cards of one suit easy
112 Four points around the centre hard
121 Waiting for the first red ball medium
124 Two numbers with no common factor hard
127 Adding until past one medium
131 How many different birthdays medium
136 Cycles of a shuffle hard
137 A boy born on a Tuesday medium
148 A series that goes the distance easy
150 A car on a quiet road easy
152 Calls that become sales medium
159 The two best players in the final medium
160 A pair or better in five cards easy
164 How far from the North Pole medium
173 A needle on a ruled floor medium
176 Two points on a circle medium
182 The red candies run out first medium
183 One extra coin medium
187 Three different faces easy
193 The Chevalier's double sixes easy
207 How far apart two dice land easy
209 Exactly half heads medium
210 A ratio between one and two easy
213 The larger of two dice easy
214 The last passenger's seat medium
218 A positive test easy
225 A sum above one, a product below 3/16 hard
227 Three boxes of coins easy
229 Which coin was it? easy
231 Only even rolls before the six hard
237 Colour runs in a shuffled deck easy
244 Closer to the centre than to the edge hard
250 The postman who likes full boxes hard
255 Counting head pairs easy
260 Spin again or not easy
261 Seated in order of age easy
264 The gap you walk into medium
268 A shared birth month easy

Other topics: random processes · order statistics · inference · regression · linear algebra · games · puzzles · all problems