# Quantitative Interview questions and answers

#### Archidamus

1. Find $$x$$ if $$x^{x^{x^{\ldots}}}=2$$
6. Calculate $$\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\ldots}}}}$$
I haven't seen this one solved the 'easy' way yet. Someone used logarithms, but that's messy.

Let $x\;=\;\sqrt{2+\sqrt{2+\sqrt{2+...}}}$
$x^2=\;2+\sqrt{2+\sqrt{2+\sqrt{2+...}}}$
$x^2=2+x$
$x^2-x+2=0$
$(x-2)(x+1)=0$
$x=2 , x=-1$ common sense check, negative radicals raises a flag. Negative 1 does not make sense.
For any value k in $\sqrt{k+\sqrt{k+\sqrt{k+...}}}$
the general solution is $\frac{1+\sqrt{1+4k}}2$

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#### Archidamus

Question: Randomly picking two points on the surface of a sphere, what is the probability that their straight-line distance is greater than the sphere's radius?

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#### VTQ

1. (BOfA, ML) There are a cup of milk and a cup of water. Take one teaspoon of milk,
put into the water cup; mix well. Take one teaspoon of the mixture in the water cup and put into the milk cup then mix well.
Which is higher: the percentage of water in the milk cup or the percentage of milk in the water cup ?

2. (Barclays, ML) $$W_t$$ is brownian motion. N is cdf of normal $$N(0,1)$$. Calculate $$E(N(W_t))$$.

3. (GS) There are 5 thieves numbered 1,..,5 trying to divide 100 gold coins using this algorithm: the number 1 will come up with a way to divide money, if there is more than 50% agreement among them about his method (including the dividing thief) then it's done. If not, then they will kill the first thief and the second thief will divide money coming up with his own method. If you are the first one, what method you will use to divide money ?

4. (SG) There is a cubic cheese 3x3x3. There is a rat eating this cheese in the following manner: it east a corner (1x1x1) of the cubic the first day. The next day, it will eat another 1x1x1 cell which has the same outer face as the one it eats the day before. Find an algorithm so that the rate can eat the center cell the last day.

5. (ML, LB, SG, Bear, DB) The today price of a certain stock is 20$. Here is an option: if the stock reaches 40$ then the payoff is 1$. Price this option. 6. (SG) $$t < T$$. W is brownian motion. Calculate $$E(W_T|W_t), E(W_t|W_T), E(W_t| |W_T| )$$ ($$W_t$$ conditioning to the absolute value of $$W_T$$) 7. (JPM) There are parallel lines with distance d lying on the same 2-D plane. There is a line segment with length l>d. Find probability that this line segment not crossing any other lines. 8. (ML) $$T_1 < T_2$$. Pricing forward-start option $$E(\frac{S_{T_2}}{S_{T_1}}-K)^{+}$$ Any Hints for Wt is brownian motion. N is cdf of normal N(0,1). Calculate E(N(Wt)). Ito's lemma? mgf ? #### Archidamus 3. (GS) There are 5 thieves numbered 1,..,5 trying to divide 100 gold coins using this algorithm: the number 1 will come up with a way to divide money, if there is more than 50% agreement among them about his method (including the dividing thief) then it's done. If not, then they will kill the first thief and the second thief will divide money coming up with his own method. If you are the first one, what method you will use to divide money ? I believe when faced with this type of problem it is okay to make assumptions(logical ones) and explain them out loud. Assumption 1: The thieves value their life more than any amount of gold Assumption 2: Thieves will only vote 'no' in favor of a death if they gain a monetary advantage, (they are thieves, not murderous sociopaths) Assumption 3: Thieves will always try to maximize their booty, and know that the other's will try to as well. Backward processing allows for the application of these assumptions cleanly. So we start with the last 2 thieves. Round 4: 4 & 5 are left, then 4 will split it 0-100 as to save his life. Round 3: 3,4&5 are left: then 3 will split it 99-1-0, as to get 4's vote as he gets a better deal than round 4. Round 2: 2,3,4&5 are left: then 2 needs to get 2 other votes. By splitting it 98-0-1-1, 4&5 each get an equal or better deal than round 3 Round 1: All are left: Then 1 needs 2 other votes. Splitting it 98-0-1-1-0, 3&4 get better or equal deals than round 2. My answer: 98-0-1-1-0 #### kalai Any Hints for Wt is brownian motion. N is cdf of normal N(0,1). Calculate E(N(Wt)). Ito's lemma? mgf ? Well One simple way is Ito lemma the expectation of the martingale term is zero and the second derivative of N is an odd function since$W$is symmetric the expectation of the bounded variation term in Ito lemma is also equale to zero so you your final result will be$N(a)$if W start at a. #### VTQ Well One simple way is Ito lemma the expectation of the martingale term is zero and the second derivative of N is an odd function since$W$is symmetric the expectation of the bounded variation term in Ito lemma is also equale to zero so you your final result will be$N(a)\$ if W start at a.
@ Kalai Thank you for the reply.

#### Archidamus

Question: A man behind a curtain roles his first die (D1), then roles a second (D2). He tells you that the second roll was higher than his first (D2>D1). What is the probability that if you roll a 3rd die, it is higher than his second die (D3>D2)? More simply, find P(D3>D2 | D2>D1), assuming all rolls are with fair 6 sided dice.

#### kalai

Well if I don't miss nothing you need to compute first P(D3>D2>D1)=(1\6)*P(the value are different)(Symmetries)=(1\6)*(5\6)*(4\6). You need on the other hand to compute P(D2>D1)=1\2P(D2 diff from D1)=1\2*5\6. and then Bays rule.

#### Rekhit

I think questions like these are something which you can work on too:

Q. What is the singles digit for 2^230?

Quickly look at the last digit of 2^n,
for n=1, 2, 3, and so on.
We only need to know the last digits of numbers. We get the units place of the multiplication table of 2 as 2, 4, 8, 6, 2, 4,.......
Thus the unitsplace is periodic and returns back after 4 numbers.

Thus 2^(320) =(2^4) 80

Thus, the units place for the number is 6
as 2^0 = 1
2^1 = 2
2^2 = 4
2^3 = 8
2^4 = 16

#### ApolloChariot

I believe when faced with this type of problem it is okay to make assumptions(logical ones) and explain them out loud.
Assumption 1: The thieves value their life more than any amount of gold
Assumption 2: Thieves will only vote 'no' in favor of a death if they gain a monetary advantage, (they are thieves, not murderous sociopaths)
Assumption 3: Thieves will always try to maximize their booty, and know that the other's will try to as well.

Backward processing allows for the application of these assumptions cleanly. So we start with the last 2 thieves.

Round 4: 4 & 5 are left, then 4 will split it 0-100 as to save his life.
Round 3: 3,4&5 are left: then 3 will split it 99-1-0, as to get 4's vote as he gets a better deal than round 4.
Round 2: 2,3,4&5 are left: then 2 needs to get 2 other votes. By splitting it 98-0-1-1, 4&5 each get an equal or better deal than round 3
Round 1: All are left: Then 1 needs 2 other votes. Splitting it 98-0-1-1-0, 3&4 get better or equal deals than round 2.

Should be 98-0-1-0-1, IF pirates would vote for a death, all other things being equal, otherwise first pirate can keep everything
Naming the pirates 1-2-3-4-5

Two pirates left, Pirate 4 gives himself all the gold, can ensure 50% agreement by voting for it himself 100-0
Three pirates left, Pirate 3 gives pirate 5 1 gold, ensuring his vote because it's a more favorable outcome for him than what would happen with two pirates left, gets 2 out of 3 votes. 99-0-1
Four pirates left, Pirate 2 gives pirate 4 1 gold, ensuring his vote. 99-0-1-0
Five pirates left, Pirate 1 gives pirates 3 and 5 1 gold each, ensuring their vote. 98-0-1-0-1

Round 4: 4 & 5 are left, then 4 will split it 0-100 as to save his life.

Doesn't need to, as he can vote for it himself and clinch the tie.

#### ApolloChariot

Question: A man behind a curtain roles his first die (D1), then roles a second (D2). He tells you that the second roll was higher than his first (D2>D1). What is the probability that if you roll a 3rd die, it is higher than his second die (D3>D2)? More simply, find P(D3>D2 | D2>D1), assuming all rolls are with fair 6 sided dice.

Seems pretty straight forward. First, compute the probability of what D2 is, given that D2 is higher than D1. 36 possibilities of the outcome of two rolls, D2 is higher in 15 of them.

5 outcomes: D2 = 6; 4 outcomes: D2 = 5; 3 outcomes: D2 = 4; 2 outcomes: D2 = 3; 1 outcome: D2 = 2

Therefore there's 5/15 chance D2 = 6, 4/15 D2 is 5, and so on

Multiply by the respective probabilities that D3 is higher....

(5/15)(0/6)+(4/15)(1/6)+(3/15)(2/6)+(2/15)(3/6)+(1/15)(4/6)

and the answer is 20/90 or 2/9, if I did my math right.

#### Fantinity

You are standing in a vast, flat desert on a moonless night. It is foggy, overcast, and raining hard. You are one mile from the only road, which is perfectly straight and infinitely long in both directions. You can’t see the road until you are right on top of it. What is your strategy for finding the road with minimum hiking in the worst case?

#### ApolloChariot

I would imagine a circle around my original location with a mile long radius. Walk in any direction to the circumference, and then walk along the circumference until I hit the road.

Wouldn't be surprised if there is a more efficient way though.

#### Bryce E

I would imagine a circle around my original location with a mile long radius. Walk in any direction to the circumference, and then walk along the circumference until I hit the road.

Wouldn't be surprised if there is a more efficient way though.
Given that it is moonless, foggy, overcast and raining hard, how could you be sure you were walking along the circumference? Footprints may not be visible and if they are they may get rained away.

Also how can you judge a 1 mile radius from your current position? If you accidentally judged 0.99 miles, you would not hit the road. Spiraling outward from your starting position might fix this but still the original vision problems might exist

#### potentialquant

How do I get latex to display on here? It only displays the source code.

#### Andy Nguyen

How do I get latex to display on here? It only displays the source code.
Click on the Insert Math f(x) icon on the editor to enter math/latex.

#### atq

From the initial list, for question 20, I've made a video recently about it (including both a simpler case solution and the generalized one)

#### ExSan

This is a great thread that has been idle for a while.

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