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Q1: What is the smallest number whose digits multiply into 216. What about 10,000? Q2: Calculate the probability of getting 3 heads after 4 coin flips. What is the probability of getting an odd number of heads for 4 flips? What about for 9? What about for N flips? Q3: What is the next date whose digits are all unique? Q4: Getting heads-tails-heads and heads-heads-tails are equiprobable after 3 coin flips. But if I keep flipping a coin, I'm more likely to get one of these combinations than the other. Why is that?
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Quantitative Trader

Interviewed at Jane Street

4.4★
May 18, 2022

Q1: What is the smallest number whose digits multiply into 216. What about 10,000? Q2: Calculate the probability of getting 3 heads after 4 coin flips. What is the probability of getting an odd number of heads for 4 flips? What about for 9? What about for N flips? Q3: What is the next date whose digits are all unique? Q4: Getting heads-tails-heads and heads-heads-tails are equiprobable after 3 coin flips. But if I keep flipping a coin, I'm more likely to get one of these combinations than the other. Why is that?

A seller is selling you a car whose value is uniformly distributed between 0 and 1000 but you don’t know the real value and you need to bid for the car. If your bid price is higher than the its real value, the deal will be done at your bid price and you can afterwards resell the car elsewhere for 1.5 times its real value. Otherwise, the deal will not be done. You can only bid once. What will be your optimal bid price?
avatar

Assistant Trader

Interviewed at Jane Street

4.4★
Feb 16, 2014

A seller is selling you a car whose value is uniformly distributed between 0 and 1000 but you don’t know the real value and you need to bid for the car. If your bid price is higher than the its real value, the deal will be done at your bid price and you can afterwards resell the car elsewhere for 1.5 times its real value. Otherwise, the deal will not be done. You can only bid once. What will be your optimal bid price?

5 pirates of different ages have a treasure of 100 gold coins. On their ship, they decide to split the coins using this scheme: The oldest pirate proposes how to share the coins, and ALL pirates (including the oldest) vote for or against it. If 50% or more of the pirates vote for it, then the coins will be shared that way. Otherwise, the pirate proposing the scheme will be thrown overboard, and the process is repeated with the pirates that remain. As pirates tend to be a bloodthirsty bunch, if a pirate would get the same number of coins if he voted for or against a proposal, he will vote against so that the pirate who proposed the plan will be thrown overboard. Assuming that all 5 pirates are intelligent, rational, greedy, and do not wish to die, (and are rather good at math for pirates) what will happen?
avatar

Trader

Interviewed at Jane Street

4.4★
Jul 17, 2013

5 pirates of different ages have a treasure of 100 gold coins. On their ship, they decide to split the coins using this scheme: The oldest pirate proposes how to share the coins, and ALL pirates (including the oldest) vote for or against it. If 50% or more of the pirates vote for it, then the coins will be shared that way. Otherwise, the pirate proposing the scheme will be thrown overboard, and the process is repeated with the pirates that remain. As pirates tend to be a bloodthirsty bunch, if a pirate would get the same number of coins if he voted for or against a proposal, he will vote against so that the pirate who proposed the plan will be thrown overboard. Assuming that all 5 pirates are intelligent, rational, greedy, and do not wish to die, (and are rather good at math for pirates) what will happen?

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