Anyway have some brain teasers they got on interviews and want to share?

### Probability Brainteasers in an S&T Interview

One user, @CUQTrader, shared a good list of trading brainteaser interview questions.

- Classic one: I can roll a die and collect the amount of money on the die, or if I don't like it, I can roll a second time and I have to pick up the die. What is my expected value?
- There is a number line 0 to 10. A random number will be randomly generated over that line and you'll get paid that much. Would you rather the numbers be on the real line or all integers?
- I like the numbers 4 and 5. I also like any number that can be added together using 4s and 5s. I.e. 9 = 4+5. 40 = 5 + 5 + 5 + 5 + 5 + 5 + 5 +5. How many number have this property from 1 to 1000?
- There is a magic island with 85 lions and 1 sheep. what makes the island magical is that all animals could live on the island only by eating grass. however, the lions would like to eat the sheep. However, when one lion eats the sheep, the sheep dies naturally, and then the lion turns into the sheep. If everyone were to act naturally how many animals would there be on the island?
- You have 100 quarters in a jar. One of the quarters is double sided (heads). You pick out a random quarter and flip it 7 times, and get all heads. what is the probability you picked the double sided quarter? Then, given that you flipped it 7 times with all heads, what is the probability that you'll get heads on the 8th flip?

Answer: Given that the expected value of one roll is 3.5, you would only re-roll for a second round if you roll is below 3.5-- So that is, you would only keep the first roll if it's 4, 5 or 6. Note that you would re-roll 50% of the time. The expected value of your first roll is therefore 5. The expected value of 2nd roll alone is 3.5. **Answer: 1/2 (3.5 + 5) = 4.25**

Answer: If 0 and 10 are included, the expected value for both is 5.

Our users discuss the answers to these questions in the comments below.

You can check out a detailed walk through of how to answer these brain teasers below.

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## Comments (36)

I got these from some well known prop shpos:

1) Classic one: I can roll a die and collect the amount of money on the die, or if i dont like it, i can roll a second time and I have to pick up the die. What is my expected value?

2) There is a number line 0 to 10. A random number will be randomly generated over that line and you'll get paid that much. Would you rather the numbers be on the real line or all integers?

3) I like the numbers 4 and 5. I also like any number that can be added together using 4s and 5s. I.e. 9 = 4+5. 40 = 5 + 5 + 5 + 5 + 5 + 5 + 5 +5. How many number have this property from 1 to 1000?

4) There is a magic island with 85 lions and 1 sheep. what makes the island magical is that all animals could live on the island only by eating grass. however, the lions would like to eat the sheep. However, when one lion eats the sheep, the sheep dies naturally, and then the lion turns into the sheep. If everyone were to act naturally how many animals would there be on the island?

6) You have 100 quarters in a jar. One of the quarters is double sided (heads). You pick out a random quarter and flip it 7 times, and get all heads. what is the probability you picked the double sided quarter? Then, given that you flipped it 7 times with all heads, what is the probability that you'll get heads on the 8th flip?

1.

Just in case.

Simpler version of this one:

"I can roll a die and collected the amount of money n the die. What's my expected value?"

1

(1/6)+2(1/6)...+6(1/6)= 1/2 (3 + 4)

= 3.5

Poster's question:

- Given that the expected value of one roll is 3.5, you would only re-roll for a second round if you roll is below 3.5-- So that is, you would only keep the first roll if it's 4, 5 or 6. Note that you would re-roll 50% of the time.

- The expected value of your first roll is therefore 5.

- The expected value of 2nd roll alone is 3.5.

- Answer: 1/2 (3.5 + 5) = 4.25

Variation of these questions:

"If I roll a die and pay you what I roll-- i.e. if I roll a 5, I'll pay you $5; how much would you pay to enter this game?"

This is the same thing as the "expected value" question-- except instead of saying "$3.5", you answer "I'll pay anything under $3.5"

who the hell can answer these on the spot during an interview

these aren't bad

Except for the last one, they are all really easy.

The last one really depends on whether you (or more accurately the interviewer) is Bayesian. I don't think it would make a good S&T question for that reason.

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anyone have answers for these?

so there are only 6 numbers that can't be mdae, so there are 994.

Therefore, the real line is preferred since it has less volatility in the distribution of results.

P(Head | 7 Heads in a row) = 1/2 --> It is independent of the first 7 flips.

Those are both incorrect. Think about using backwards induction for problem 4. For problem 5, i think you missed that one of the coins is double sided.

you are an idiot.

By Bayes theorem,

P( you picked double sided | you flipped 7 heads in a row) = P (7 heads in a row | double sided) P (double sided) / P(7 heads in a row)

P(7 heads in a row | double sided) = 1

P(double sided) = 1/100

P(7 heads in a row) = (99/100)*(1/128) + (1/100)

Hence P (double sided) = 1/ (99/128 + 1) = 128/227

(hope I got the arithmetic right, but I think my methodology is correct)

Thus, the chance that the next flip will be heads is (128/227 + 99/454)

If you don't understand this, pls don't bother to reply and tell me I am wrong.

@ silver94

This is correct.

And these aren't 'brain teasers' other than the sheep one.

They're basic probability questions to see if you can do basic math.

This is correct, but you could just use the events as they are given.

You want

P(heads on 8th | flipped 7 heads already)

This is the same as

P(heads on 8th and flip 7 heads) / P(flip 7 heads)

Numerator is = 1/100

1 + 99/100 * (1/2)^8 (taking the two cases of fake coin and real coin)1 + 99/100 * (1/2)^7Denominator is = 1/100

So its (2^8+99)/(2^8+99*2) = 355/454

Are you really expected to calculate this during the interview?? Are you given pencil and paper, and how much time are you given?

Also, are there any resources that have questions encountered in interviews with places like DRW/Jane St? The ones on WSO from other threads are pretty easy, except for maybe calculation.

This is what I am interested in, it seems pretty hard if you are not allowed to write it down.

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Hey, answer for the lion one was 84 lions and 1 sheep. I was only able to get this because I saw the pirate gold question about a day before. Check that one out and hopefully you'll see how to solve it.

Yea, 84 lions and 1 sheep assuming that the lions saw the 85th lion eat a sheep and turn into a sheep. They logically get scared and go back to living on grass.

Bankingrox: thats not quite the reasoning for why 84 lions and 1 sheep. I guess you should assume that lions know what happens.

If you work backwards, imagine there is 1 lion and 1 sheep, what would happen? Then what if 2 lions and 1 sheep? etc.

you are allowed to write it down. and except for the sheep-lion question, pretty standard ones. The first question I got asked in three different interviews. You have to be able to solve that one. The last one is a bit long, but by Bayes theorem, it is just calculation-wise long, but concept is simple if you understand it.

im still curious on the second one...can anyone clarify?

the volatility thing is true, but i was also thinking in the direction of rounding. given that you round up to whole integer at x.500000, theres a slightly higher probability of hitting the higher integer and thus its preferrable?

Why would you round up? You can get paid in dollars and coins, not just whole dollar amounts. If the problem said I only have dollar bills and no coins then you would be right, but you have to assume coins are involved so the rounding only occurs in the 3 decimal place.

For some reason, I was only given a forward rate question. The guy asked one year at 1%, or two at 1.5%. What is the rate next year. Besides that, nothing close to a brain teaser.

m.c.trader, read the question again I'm pretty sure the choice is between integers and real numbers. has nothing to do with dollars or coins

2) There is a number line 0 to 10. A random number will be randomly generated over that line and you'll get paid that much. Would you rather the numbers be on the real line or all integers?

^ Original post: right back atcha

It's all about getting paid, like most brain teasers, they will all take prob/stats and relate it back to money.

So my answer: You can't assume that everything is going to be rounded if it's on the real line with an int > 5 in the 1st decimal place.

You have to assume that you can be paid, for example, $3.32(12565436) or $3.56(345456234) as opposed to the dollar amount of the nearest integer, in this case $3.00 or $4.00

Of course the expected value is the same, that's the point, they want to see if you can consider volatility of the outcomes.

Have you ever seen a day trend for a really illiquid stock? looks like 6 or 7 straight lines, while a high volume stock has a day trend that is a continuos line? You want the continuos line, not the extremely volatile stock that swings at the drop of a hat.

The same goes with this brain teaser, you want the more continuos pay out line as opposed to the volatile line that could go from 9 dollars to 1 dollar in one iteration.

^ To that end, i realize you don't always want to pick the least volatile stock, more risk equals more reward for some trading strategies, but in generalities the answer you want to give the real line when in an interview.

i guess that depends on how they produce the random integers....

i definitely agree with you on the volatility point. although the way that question is phrased implies there is only one trial so i assumed thats irrelevant. thats why i wasnt sure thats the answer they are looking for

For #4, I think the question is stated very poorly, especially as it states "if everyone were to act naturally...". That implies some understanding of what "natural" behavior is for a lion, and since all we're given in the way of animal psychology is "the lions would like to eat the sheep". No indication of the lion's ability to reason, how observant they are, whether or not they act in the interest of self-preservation or are prone to binging on sheep at all cost. As the problem is stated, I would say that there is not enough information to solve the problem, but that one likely answer is that the island would reach equilibrium with only 1 sheep left.

On the other hand, doing a quick search, I see other versions of this same problems (such as http://theweeklyriddle.blogspot.com/2010/01/lions-...), where they are careful to state that the lions are "perfectly rational" and then state the rules of their behavior. Although even that web site has a typo near the bottom of the conclusion, with the situation for "even" lions and "odd" lions being reversed. If you redefine the problem carefully, the answer is that you end up with 84 lions and one sheep, but not because it took one lion/sheep transformation for the lions to "figure it out". By the way, the version of the problem I linked to doesn't actually state that the lions are aware of the rules of the island. It's all very well to say that they're "perfectly rational", but they also have to know the rules of the island vis a vis lions turning into lambs when they eat them, etc. And they also have to know that the other lions are "perfectly rational" and that THEY know the rules of the island, and that they KNOW that the other lions all know that the other lions all know that... It's difficult to really ask the question you're trying to ask! :)

Apologies in reviving an old thread, but according to problem 6 (the double-headed quarter question)... how would you know to use Bayes Theorem?? In comparison to just using regular statistics?

Anyone?

You do it using Bayes. Look at Silver's answer

What materials do you guys recommend to prep for S&T interviews for the mental math/brain teaser portion?(Originally Posted: 02/07/2014)Would really appreciate the input. Thanks guys.

Heard on the Street by Crack is the standard... there should be pdfs floating around if you don't want to purchase the actual book

Really appreciate it

+1 Heard on the Street... I found this was also quite good - http://www.decal.org/file/2945, would also recommend brushing up on stats (CFA curriculum's Level 1 quant section is good for this) /probability theory (CFA good again /calc (http://www.amazon.com/Introduction-Mathematics-Financial-Derivatives-Academic/dp/0125153929). Using those resources (+ others) I landed a FT offer recently.

the idea is more about knowing your thought process. try this, helped me a lot during my interviews.

We go to war to learn about geography

2) Both versions have the same expectation value, but the variance for the integers is 10 which is higher than in the rationals case (8.33)

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