Compilation of Jane Street interview questions

[QuantI]

Hey guys, just wanted to share this with you:

http://canarywharfian.co.uk/threads/jane-street-trader-interview-questions.61/
http://canarywharfian.co.uk/attachments/jane-street-pdf.6/

 

[Earl Chua]

Question, re: the problem with a 30-sided and 20-sided die.

You have a thirty-sided and I have a twenty-sided die. We roll both of them and whoever gets the higher roll wins. If we roll the same amount I win. The loser pays the dollar amount of the winning roll.

B) What is the expected value of this game for you?

The answer given is 10.2 but I keep getting something else. I tried many different ways, including the one prescribed in the solution, but I get the same value that's not 10.2.

 

[euroazn]

Question, re: the problem with a 30-sided and 20-sided die.

You have a thirty-sided and I have a twenty-sided die. We roll both of them and whoever gets the higher roll wins. If we roll the same amount I win. The loser pays the dollar amount of the winning roll.

B) What is the expected value of this game for you?

The answer given is 10.2 but I keep getting something else. I tried many different ways, including the one prescribed in the solution, but I get the same value that's not 10.2

Do you get 8.15?

 

[Earl Chua]

Do you get 8.15?

Yes

 

[euroazn]

Yes

I think that's right, but I didn't spend much time on the problem.

 

[Dave Lee]

I got 8.15 as well.
Just have a question, are we allowed to use pencil and paper to do this calculation during interview?

 

[euroazn]

I got 8.15 as well.
Just have a question, are we allowed to use pencil and paper to do this calculation during interview?

Some questions you can, some you can't.
I would say that for this question you should be able to do it in your head. Just split it into the case where the 30-die rolls >20 and the other case which is equivalent to two 20-dice. Then exploit symmetry.

Their solutions are pretty terrible lol

 

[CasanovaJ]

If I had to come up with something quickly on this, I'd get 12...... there is 1/3 chance that I roll something from 21-30 which is a guaranteed win, the EV in that case is 25.5, in the 2/3 chance that I get something from 1-20 there is a 50% chance of a win (since in that case we're both going to get something random from 1-20, expected roll for both is 10.5, and there is equal chance of each beating the other), so overall EV is 1/3 * 25.5 (rolling 21-30) + 1/3 * 10.5 (rolling 1-20 and winning) + 1/3 * 0 (rolling 1-20 and losing) = 12. Where is my thinking wrong?

Edit-- forgot about the negative EV in the case of losing... in that case, though, now I am at 1/3 * 25.5 (rolling 21-30) + 1/3 * 10.5 (rolling 1-20 and winning) + 1/3 * -10.5 (rolling 1-20 and losing) = 8.5, not 8.15

Additional edit-- I guess I probably lose a bit from the "tie goes to you" thing, but how my loss from that would equal 0.35 (in a way that one could calculate in their head) is tough... I'd have to draw a grid with all the possible pairs and tally up the results...

Edit #3-- OK, yeah, now I'm getting 8.15... It took building a 20x20 grid in Excel, though, (EV of -.525, -0.525 * 2/3 = -0.35) which I don't think Jane Street lets you do. How could this possibly have been done mentally with no calculator/computer?

Just split it into the case where the 30-die rolls >20 and the other case which is equivalent to two 20-dice. Then exploit symmetry.

What symmetry?

 

[euroazn]

If I had to come up with something quickly on this, I'd get 12...... there is 1/3 chance that I roll something from 21-30 which is a guaranteed win, the EV in that case is 25.5, in the 2/3 chance that I get something from 1-20 there is a 50% chance of a win (since in that case we're both going to get something random from 1-20, expected roll for both is 10.5, and there is equal chance of each beating the other), so overall EV is 1/3 * 25.5 (rolling 21-30) + 1/3 * 10.5 (rolling 1-20 and winning) + 1/3 * 0 (rolling 1-20 and losing) = 12. Where is my thinking wrong?

Edit-- forgot about the negative EV in the case of losing... in that case, though, now I am at 1/3 * 25.5 (rolling 21-30) + 1/3 * 10.5 (rolling 1-20 and winning) + 1/3 * -10.5 (rolling 1-20 and losing) = 8.5, not 8.15

Additional edit-- I guess I probably lose a bit from the "tie goes to you" thing, but how my loss from that would equal 0.35 (in a way that one could calculate in their head) is tough... I'd have to draw a grid with all the possible pairs and tally up the results...

Edit #3-- OK, yeah, now I'm getting 8.15... It took building a 20x20 grid in Excel, though, (EV of -.525, -0.525 * 2/3 = -0.35) which I don't think Jane Street lets you do. How could this possibly have been done mentally with no calculator/computer?


What symmetry?

First of all, not quite:

in the 2/3 chance that I get something from 1-20 there is a 50% chance of a win

Because remember, if you tie, you lose.

As for the symmetry, I mean in the 20 die vs 20 die game, the ev you get if you're higher is equal to negative of the ev you get if you're lower. The only net contribution of the 20 die vs 20 die game is negative EV you get from a tie, e.g 1/20(10.5)

Consequently EV from the entire game is 25.5*1/3-2/3*1/20*10.5 which is easily computable in your head to be 8.15

Makes sense?

 

[CasanovaJ]

First of all, not quite:

Because remember, if you tie, you lose.

As for the symmetry, I mean in the 20 die vs 20 die game, the ev you get if you're higher is equal to negative of the ev you get if you're lower. The only net contribution of the 20 die vs 20 die game is negative EV you get from a tie, e.g 1/20(10.5)

Consequently EV from the entire game is 25.5*1/3-2/3*1/20*10.5 which is easily computable in your head to be 8.15

Makes sense?

Yeah, I'd already caught on with the tie thing (see my various "edits" above ) but I'm understanding the symmetry thing now... Good call!

 

[Sixth Symphony]

Necromancing over half a decade later: Did anyone manage to save a copy of the PDF linked? Unfortunately the Wayback Machine did not save it...

 

[stochastic_descent]

Same for me, could please someone post PDF of those interview questions? Thanks!