Secret Slovak Crypto Trading Quants Interview questions

wincent_quant_interview_questions_2022.md

Wincent Quants Interview Test (2022)

I came over this test, sticking it here so that anybody can have some fun. To provide a solution or request one, please leave a comment. Wincent likely already changed the questions, so the test is just a good training for any Quant role.

1. Even heads

You toss a fair coin 100 times. What is the probability that the number of heads you get will be divisible by 2. Why?

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2. A lucky dice

You roll a fair 6-sided dice and you receive a $1.00 for every dot landed. What is your expected profit?

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3. A lucky re-roll dice

You play the previous game, but you can re-roll the dice once if you don't like the result. What is the expected profit now?

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4. Ping-pong

You play ping-pong and the current score is 10:10. The first one to gain a lead by 2 points wins. Your chance to score a point is 40%, for your opponent it's 60%. What is your chance of winning?

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5. Tulip-27

A deadly virus called Tulip-27 emerged in Quantaria. Fortunately, there is a test with following properties:

• If an infected person is tested, there is a 30% chance that the test will come up as (false) negative.
• If a healthy person is tested, there is a 10% chance that the test will come up as (false) positive The king has decided to test all of his 50 million subjects for the disease. As a result, 16% of the population had a positive test. What is the true number of infected people?

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6. Russian roulette

You are playing a Russian Roulette. There are precisely two bullets in neighboring chambers of the six shooter revolver. The first person spins the barrell, pulls the trigger and the gun does not fire. You are next. You just pull the trigger without spinning. What is your chance of dying? What would the chance be if you would spin?

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7. Treasures

There are 4 treasure chests. One of them contains $100, the other three are empty. If you pay a fixed fee of X you can open any of the chests and take everything you find there. If you find an empty chest, you can pay again to open the second, third and fourth, though you always have to pay X for each chest you open. What is a fair X?

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8. Male heir

In Quantaria, parents breed kids until their first boy is born. Some families have a single boy, some families have 15 daughters and one boy. Assuming there is a 50% chance for every single pregnancy to end up as boy or girl, what is the ratio of boys to girls in Quantaria?

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9. Cube

You start in one of the corners of the cube. Then you randomly pick one nearby edge and move to its end. You then repeat this process indefinitely. What is the expected number of moves you will make before you reach the opposite corner of the cube?

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10. Stick

You randomly break a stick into two pieces. What is the expected length of the smaller piece?

***Spoilers alert: Answers below

1. 1/2 -
   works for any arbitrary number of tosses. If n-1 tosses has even number of heads we want tail on the nth toss, and vise versa. So either way the nth toss will determine the result, and its 50/50 chance.

2. $3.5
   Expected value of dice = 1/6(1+2+3+4+5+6)=3.5

3. $4.25
   Reroll second time if first roll < expected value (i.e., 3.5). That is, keep first roll if value = 4,5,6. E[V]=0.5*(1/64+1/65+1/66)+0.53.5=4.25

4. 4/13
   After every two points, 0.4*0.4 chance you win, 0.6*0.6 chance you lose. 2*0.4*0.6 chance the game remains tied, in which case your probability of winning then is the same as start state. Setting equation x = 0.4*0.4*1+0.6*0.6*0+2*0.4*0.6*x and solving for x we get 4/13

5. 5 Million
   Let x be the percentage of people infected, and 1-x be the percentage of healthy people. Setting equation 0.7x+0.1*(1-x) = 0.16 and solving for x gives us 0.1. 0.1*50 million = 5 million.

6. Chance of dying w/o spinning = 1/4, w/ spinning = 1/3
   w/ spinning: 2 bullets out of 6 chambers = 2/6 = 1/3 (assuming you would die 100% if bullet fires lol)
   w/o spinning: you know the previous person fired one of the 4 empty chambers, as long as its not the one right before the bullet you are safe.

7. $40
   There is 1/4 chance to find treasury on the first chance, second chance, etc. The expected amount you pay would then be equal to (x+2x+3x+4x)/4 = 10x/4. For fair X, make that equal to 100 and we get 40.

8. 1:1
   Every family has exactly one boy. The expected number of girls in a family is a geometric distribution w/ success probability 0.5, which has an expected value of (1-p)/p, where p=0.5, equating to 1.

9. 10
   Denote the vertexes in 3-D space (x, y, z) where each of x,y,z is 0 or 1. Say we start at (0, 0, 0) and want to move to (1, 1, 1). Each move, we can switch exactly one of x, y, or z.
   Let f(x,y,z) denote the expected number of steps to reach vertex (1,1,1). We see that f(1,1,1)=0 (already there). Taking f(0,1,1) as an example, after one move, it has equal chance of being at each one of the three vertexes (1,1,1), (0,0,1) or (0,1,0). That is, f(0,1,1)=1 + 1/3(f(1,1,1) + f(0,0,1) + f(0,1,0)). By symmetry, we see that f(0,1,1)=f(1,0,1)=f(0,0,1). Applying such logic to each of the other vertexes and solving for the system of equations, we get that f(0,0,0)=10.

10. 1/4 of the length of the original stick
    Suppose the position of breaking x is uniformly distributed over [0,1]. shorter piece = min(x, 1-x) = x if x <=0.5, = 1 - x if x > 0.5.
    Imagine the unit square. Shade in the area below y=x for x<=0.5 and the area below y=1-x for x>0.5. We get a triangle with base 1 and height 0.5, and thus area = 0.25. The apply that ratio to the length of the stick.

Please, put a disclaimer there, so whoever keeps reading won't go into the solutions by mistake. Other than that: 🍻