Investment Banking puzzles, solved step by step

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001You have two empty bowls, 50 white balls and 50 black balls. Split all 100 balls between the bowls any way you like. A bowl is then picked at random and one ball drawn from it. How do you maximise the chance of drawing white?ProbabilityHardDeutsche BankMumbai · 2024
Try it first
Before you work it: what is the best chance of white you can get?
Show the worked solution
Put one white ball alone in bowl A and the other 99 balls in bowl B. Bowl A gives white every time. Bowl B gives white 49 times in 99. Each bowl is picked half the time, so the chance of white is 0.5 x 1 + 0.5 x 49/99, about 74.7%, against 50% for an even split.
Why does the number of balls in a bowl not matter?
Think of two queues at a ticket counter, one with a single person and one with ninety nine. If a coin decides which queue gets served, the lone person gets served half the time. The bowls work the same way. The bowl is chosen with even odds, so a bowl holding one ball carries exactly as much weight as a bowl holding ninety nine. That is the lever: you can make one half of the draw certain at the cost of a single white ball.
One white ball alone makes bowl A a certain white. Bowl B keeps 49 white and 50 black, so white comes up 49.5% of the time there, and the average across the two bowls is 74.7%, well above the 50% an even split gives.
How do you prove no other split beats it?
Bowl A can never beat 100%, and one ball is the cheapest way to get there. Every extra white ball you move into bowl A is a white ball taken out of bowl B, which lowers bowl B's odds without raising bowl A's. Adding black balls to bowl A only hurts it. So the best bowl A is one white ball, and bowl B's best case is whatever is left: 49 white out of 99.
The relationship
$P(\text{white}) = \tfrac{1}{2}\cdot\frac{1}{1} + \tfrac{1}{2}\cdot\frac{49}{99} \approx 0.747$
1/2 the chance each bowl is picked
1/1 the chance of white from bowl A, one white ball
49/99 the chance of white from bowl B, 49 white in 99 balls
What it says in wordsAverage the white odds of the two bowls, because each bowl is equally likely to be picked.
Say the 74.7% figure out loud, then say the general rule: with n white and n black balls, the answer is one half plus one half of (n minus 1) over (2n minus 1), which creeps towards 75% as n grows. Interviewers like hearing that you see the limit.
Where candidates lose it
Most candidates answer 50% and stop, because it feels as if a fixed pool of balls must give a fixed chance. The interviewer is testing whether you notice that the bowl is picked first, with even odds, whatever it holds.
The second trap is getting 74.7% and then being unable to say why it cannot be beaten. Have the one-line proof ready: a white ball moved into bowl A beyond the first one is wasted there and missed in bowl B.
What the interviewer asks next
What if the bowl is picked with probability proportional to how many balls it holds?
What if there are three bowls?
Now I pay you 100 if you draw white. What would you pay to play?
Asked at Deutsche Bank, Equity Capital Markets, Mumbai, 2024 (Wall Street Oasis): The most difficult question was the brain teaser. You have two empty bowls and 100 balls

1/2	the chance each bowl is picked
1/1	the chance of white from bowl A, one white ball
49/99	the chance of white from bowl B, 49 white in 99 balls

Try it first

Why does the number of balls in a bowl not matter?

How do you prove no other split beats it?

Where candidates lose it

What the interviewer asks next