Assistant trader Interview Questions

P(A|HH) = 0 because after a sequence of consecutive heads, you can no longer achieve HTT. The moment you get a tail, you will have the sequence HHT. This the reason HHT is more likely to occur first than HTT.

HHT is more likely to appear first than HTT. The probability of HHT appearing first is 2/3 and thus the probability of HTT appearing first is 1/3. Indeed, both sequences need H first. Once H appeared, probability of HHT is 1/2 (b/c all you need is one H), and probability of HTT is 1/4 (b/c you need TT). Thus HHT is twice is likely to appear first. So, if the probability that HTT appears first is x, then the probability that HHT appears first is 2x. Since these are disjoint and together exhaust the whole probability space, x+2x=1. Therefore x=1/3.

You guys seem to be mixing order being relevant and order being irrelevant. If order is relevant (meaning HHT is not the same as HTH) then this has a 1/8 of occuring in the first 3 tosses. Also HTT has a 1/8 chance of occurring in the first 3 tosses, making them equally likely. Now, if order is not relevant. (so HHT and THH are the same), then this has a (3 choose 2) * (1/8) probability of happening in the first 3 tosses. The same goes for HTT (which would be the same as THT etc and others) so this has a (3 choose 2) * 1/8 probability of happening in the first 3 tosses as well. Either way they come out to being equally likely, please comment on my mistake if I am doing something wrong.

Ending up with HHT more likely (with probabilty 2/3).

HHT is more likely (2/3) probability. People with wrong answers: Did you not Monte Carlo this? It takes 5 minutes to write a program, and you can then easily see that 2/3 is correct empirically.

I don't get it. Shouldn't P{A|HH} = P{A} in the same sense that P{A|HTH} = P{A} from both HH and HTH we have get the first H from HTT and so it should be P{A|HH} = P{A|HTH} = P{A} Am i wrong?

sorry, i meant: I don't get it. Shouldn't P{A|HH} = P{A|H} in the same sense that P{A|HTH} = P{A|H} from both HH and HTH we have get the first H from HTT and so it should be P{A|HH} = P{A|HTH} = P{A|H} Am i wrong?

Above link is the best solution I have seen for this problem http://dicedcoins.wordpress.com/2012/07/19/flip-hhh-before-htt/

Apologies, Below*

Here's my answer. Let x = probability of winning after no heads (or a tail). y=probability after just one heads. z=probability after two heads. w=probability after HT. Thus x=(1/2)x+(1/2)y, y=(1/2)z+(1/2)w, z=1/2 + (1/2)z, w=(1/2)y. Therefore, z=1, y=2/3, w=1/3, x=2/3. We wanted x at the beginning, so it is 2/3 that HHT comes up first.

Think of HHT as H(HT) and HTT as (HT)T For every occurrence of (HT) in a sequence of flips, there is a 1/2 chance an H occurred before the (HT) and a 1/2*1/2 chance a T occurred after. Thus, HHT is 2x more likely than HTT. HHT is first 2/3 of the time.

With 1 coin, flip three times and HHH=1,HHT=2, HTH=3,HTT=4, THH=5, THT=6. If the first twice is TT, then abandon the first T and start with the second T.

The candidate and William have problems. I think Ey's solution works. For the candidate's solution, you will get way more 5s and 6s, since fully 1/2 of the results will be either a 5 or a 6. Similarly, for William's solution, if you toss out results starting with HH, then you end up with ANY result that starts with a tail, and only half the results that start with a H. That means you will end up with more 3s, 4s, 5s, and 6s than you should. I think by simply changing what you throw out you could fix William's solution. If you threw out all HHH and all TTT you would be closer, but a solution where you don't toss out any data is still better.

Nope. I'm a moron. Can't edit earilier post, but Ey, sorry, I gave you too much credit. Can't count HT and TH as the same thing because you get too many of them. No idea why I didn't see that 10 minutes ago. You can't map 4 coin results to three numbers fairly. You will end up with too many 2s and 5s and way too many 6s with your solution. Now we have no valid answers. I think someone proved it can't be done elsewhere. Interesting problem though.

Toss 1: Heads=even, start with 2, tails=odd, start with 1 Toss 2: Heads=0, tails=2, add to total Toss 3: Heads=0, tails=2, add to total

It cannot be done. The odds of any set of coin flips is always a power of 2, and there is no power of 2 for which 6 is a factor, because 6 factors to 2 * 3 and both are prime numbers.

Eric, you're right, I thought my solution worked as well. My idea was that we do 2 independent 2-coin flips. Each 2 coin flip has the following possible outcome: HH, TT, HT(or TH, same thing). When combining the outcomes of the 2 independent 2 coin flips this is what we get: (1) HH (2) HH (1) HH (2) HT (1) HH (2) TT (1) TT (2) HH (1) TT (2) HT (1) TT (2) TT (1) HT (2) HT (1) HT (2) HH (1) HT (2) TT we get 9 possible outcomes, even if we set for example outcomes (1) HH (2) TT to be equivalent to (1) TT (2) HH, and similarly with others to get 6 outcomes, it won't work...

HHH, TTT, start the process again. Assign the rest values of dice. HHT = 1 HTH = 2 HTT = 3 THH = 4 THT = 5 TTH = 6

Geez, they should really allow for nested responses for better discussions. I agree that with the approach first stated by William (except he had a typo, "HHT or HHT appears" should read "HHH or HHT appears" he was just referring to first two (or any two of the eight combination that one chooses not to assign values to). The way that "Do three flips" describes is clearer. Other approaches are problematic in that the chances of getting 1-6 are disproportionate leading to an unfair dice (for reasons that others stated). "Binary Thinkers" approach is also incorrect as for similar reasons. there 8 possible accounts (3 flips 2^3 = 8). Although it's a unique way to account for all the numbers 1 through 6... 3's are and 4's are precisely twice as likely to occur (2/8 or 1/4 chance) vs the other numbers (1/8 chance).

You may ask for a probability of THHHHH if we get a consequent combination with 5 heads and one tail. Here are the 6 possible combinations: THHHHH HTHHHH HHTHHH HHHTHH HHHHTH HHHHHT so the probability of each is 1/6

Not a precise solution, but good enough for many purposes: Flip the coin a large number of times, interpret the result as number in base 2, and take it modulo 6. Some of the six numbers will be slightly disadvantaged, but you can make that difference arbitrarily small by increasing the number of coin flips.

can someone explain why it is the case mathematically?? I thought your expected payoff is (0+1000)/2 x 1.5=750.. so if you bid 500. you will get expected 250 profit...

Anon, If you bid 500 and it is worth 1, you only get paid $2 so you lose 498. $0 is correct. Generalize this for 1,2,5,10 maybe and find the EV of all those, or just 1,2,10, and then you will be able to "guess" that it isn't getting more and more negative and you should bid 0

you should bet 1, that is the minimum in the box, so you will win 0 or .5

if you bet one you'll never win. if you bet 500, you're missing out if value(box) >500 and <750 since those are +EV too. believe the answer is 749.

The answer is bid 0 dollars. Soln: If you bid X dollar, you get the box if if it contains Y dollars, where Y < X. Or you nothing, which doesn't hurt anyway. Now think of possible values of Y in PAIRS, (0, X), (1, X-1), (2, X-2)... in each pair (a, b), expected payoff is P(Y being a or b) * E(payoff given Y = a or b). but the second term E(payoff given Y = a or b) = 1/2 (3/2 a - X) + 1/2 (3/2 b - X) = 3/4 (a + b) - X = (-1/4) X, which is always negative. If we sum over all possible pairs, we should get a negative number. Ofocz you need to handle the edge case, where X is small and the pairing does not really work.

I think that since we are dealing with expected profit, we should ignore the case when n (our bid) is less than X (cash in the box) as nothing happens in this case. So we create a new distribution where X is uniformly distributed on [0, n]. Thus, E(Profit) = E(1.5X - n) = 1.5E(X) - n = 1.5*n*(n+1)/(2*(n+1)) - n = 0.75n - n = -0.25n <= 0. So the expected profit is maximised (=0) if n=0.

Sorry, guys. I seem to have made a mistake in my solution above. Let X = cash value, n = our bid price. Then Profit = 1.5X - n. We want to find E(1.5X - n) = 1.5*E(X) - E(n). Now in order to find E(X), we notice that that X is uniformly distributed. So every outcome has a probability 1/1001. Also we only need to sum up over the values from 0 to n. Hence, E(X) = n(n+1)/(2*1001). For E(n), we see that we will pay the amount n with probability (n+1)/1001 and 0 otherwise. So E(n) = n(n+1)/1001. Therefore, E(Profit) = 1.5*n(n+1)/(2*1001) - n(n+1)/1001 = -0.25*n(n+1)/1001<=0. In order to maximize profit, we should bid zero.

The answer is 0 if you ask me and here is the reason. let V be the value of the box and let x be your bid then you profit P is P=(1.5V-x)1_{V<=x}. we take expected value of this and use conditional expectation to get E(P)=E(1.5v-x|v<=x)p(v

Answer is 2. The key is that the amount of cash X is uniform on (1,100) rather than (0,100). If we bid y money, then are expected value of profit P is E[P | bid y]=(\int_1^{y}(1.5-y)dx)/99=-(y^2-4y+3)/(4*99). Here the limits of integration are because P will be 0 if X is smaller than y, and we will get profit 1.5x-y if X=x and x<=y. Solving the maximum of this function gives y=1, with an expected profit of 1/(4*99)=0.0025. If the value was in fact from (0,100), then our expected value for any y would just be -y^2/(4*99) so we wouldn't want to play that.

11/3. E(x) = .75*3 + .25*E(X+2).

The probability of a six sided die is 1/6, while the probability of a coin flip is 1/2, thus, in order to simulate a six sided die the coin must have the same probability, so 1/2 has to become 1/6. From there, you can write the equation1/2^x=1/6, or to make it simpler, 2^x=6. Without using guess and check, you can rewrite the equation to log base 2 6=x (or log2(6)=x), and from there solve: log(6)/log(2), which comes out to 2.58496, or 2.585 (2^2.585=6.000). So, to answer your question, the coin would have to flip 2.585 in order to simulate the flip of a coin. x=2.585

3 flips. The coin gives 1 out of 2 possibilities per flip. 3 flips will give you 6 possible combinations like the 6 sided die (1 heads, 2 tails, 2 heads, 1 tail, etc.).

It is easier to look at it as almost 3 flips set up as a binary number. It would be perfect if you were representing an 8 sided die, but since you want only 6 possible results and 2 flips is too little and 3 flips is too big you have to adjust your fomula as such: 0=tails 1=heads 000 = Null this means you scratch these results and flip again three more times. 001 = 1 010 = 2 011 = 3 100 =4 101 = 5 110 = 6 111 = is another Null result Of course the Nulls can be made to be anything as are the numbers it is all relative. This give you a 1 in 6 chance everytime.

Wsc is on the right track. You have to flip 3 coins at least, but the probability of a null result is 1/4 every time, so you would need to flip 3 more coins. This is a geometric sequence 3+3*1/4+3*(1/4)^2 etc, so the answer is 3/(1-1/4) = 4 expected coin flips. 4

The correct answer is 11/3, as written by "Someone" on July 9, 2011. Both William's and P's answers are close, but not quite accurate, because they assumed that it takes 3 flips to discard a null result. Instead, it only takes two flips to discard a null result (HHT or HHH) since we don't have to flip the third coin if the first two are already heads (HH). That's why the answer is slightly less than 4. Indeed, note that we have a 3/4 chance of being able to get one of the 6 good results (HTH, HTT, THH, THT, TTH, TTT) that yields a random number from 1 to 6, and a 1/4 chance of having to reflip the coins (HHH or HHT). The key is that for those two results to be discarded, we only needed to flip 2 coins to determine that! There's no need to flip a 3rd coin if the first two were already HH (it'd be a waste of a throw!). Hence, 3/4 of the time, the number of flips required is three and 1/4 of the time, the expected number of flips required is 2+E[X]: E[X] = 3/4*3 + 1/4*(2+E[X]). Alternatively, we can flip the coin twice, and then note that 3/4 of the time (for HT,TH,or TT), we only flip one more coin, and 1/4 of the time (for HH), we have to re-flip, so E[X] = 2 + (3/4*1+1/4*E[X]) Either way, we have E[X] = 11/4 + 1/4*E[X], which yields the desired result, E(x) = 11/3.

4.5

It's 11/3. Say you want HHH and TTT to be your rolls to start over so that the last roll is 50-50 to be H or T and you'll only have to roll twice more since there will be no bias. Roll your first roll, it's H or T. After that there is a probability of 1/4 that you'll have to 2 times again. So there a 3/4 chance you won't have to roll again. So using the equation 1/p to find EN we get 1+2(4/3)=1+8/3=11/3

You can't flip a die because you only have a coin. DUH

probability shooting star in half hour = x probability of NOT seeing shooting star in 1 hour = (1-x)^2 probability of seeing shooting star in 1 hour = 1 - (1-x)^2 so we solve 1 - (1-x)^2 = 0.91 for x

According to the question, suppose we have a random variable X that is the time when we see a shooting star, and let F(x) be the CDF of X. Hence, F(now+1h)-F(now)=0.91. And this is unfortunately the only thing we know about F(x). In other words, any valid F(x) satisfying such requirement could be the CDF of X. Therefore, we have no idea of the value of F(now+0.5h)-F(now) because F(now+0.5) could be any value between F(now) and F(now)+0,91. Answer: unknown result depending on the distribution.

The second answer is incorrect. Take an extreme case as an example, if the probability of seeing a shooting star in 1 hour is 1, then the root of the function would be 1, which means that the probability of seeing a shooting star in the first 30 minutes is 1. An interesting question followed by that would be: what about in the first 15 (second 15) mins? According to this method, we can again conclude that the probability of having a shooting star in the first 15mins is still 1. And this can go on for ever, and result in the conclusion that at any time interval, P(shooting_Star) would be 1 which is obviously mistaken.

let p= probability to see the star in half hour, (1-p)=won't see, (1-p)^2=can't see a star in an hour=1-0.91, solve we have p=0.7

The answer should be 0.7 but most of the explanations here make no sense. It is reasonable to assume a exponential distribution here and solve the probability.

P(no star in an hr) = 1 - 0.91 = 0.09 P(no star in half an hour)^2 = 0.09 P(no star in half an hour) = 0.3 P(star in half an hour) =1 - 0.3 = 0.7 = 70%

Probability is 30%. The arrival of star follows a Poisson process. Seeing K arrivals in time T is P = (lambda*T)^K * exp(-lambda*T)/K!. Now we know there has been no arrival in 1 hour, T = 1, K = 0. P = exp(-lambda) = 1-0.91 = 0.09. Now T = 0.5 (half an hour), P = exp(-lambda*1/2) = sqrt(0.09) = 0.3. The probability of seeing a star is 1-0.3 = 0.7, or 70%.

I would argue for the 91/2% = 45.5% chance of seeing the star. The wording of the problem implies this is not a poisson process. Rather, I would interpret as the weather channel telling residents in an area that there is a 91% chance of seeing the shooting star between 7pm and 8pm (this is an unusual occurrence). Hence, with 9% chance you can't see a star in the next half hour. Then, in the 91% of the time where there is a shooting star in the next hour, one could make the reasonable assumption that the time it comes is uniformly distributed in the hour. So, 45.5%.

Like, think about it in the real world. It is kind of unreasonable to model seeing shooting stars as a Poisson process with parameter .91 stars per hour.

The answer is 7 because the expected value of rolling a single dice is 3.5 ((1/6 * 1)+(1/6*2) + (1/6*3) + (1/6*4) + (1/6*5) + (1/6*6)), so for two dice just multiply 3.5 * 2 = 7

The answer is 7 because the expected value of rolling a single dice is 3.5 ((1/6 * 1)+(1/6*2) + (1/6*3) + (1/6*4) + (1/6*5) + (1/6*6)), so for two dice just multiply 3.5 * 2 = 7

The answer is 7 because the expected value of rolling a single dice is 3.5 ((1/6 * 1)+(1/6*2) + (1/6*3) + (1/6*4) + (1/6*5) + (1/6*6)), so for two dice just multiply 3.5 * 2 = 7

The answer is 7 because the expected value of rolling a single dice is 3.5 ((1/6 * 1)+(1/6*2) + (1/6*3) + (1/6*4) + (1/6*5) + (1/6*6)), so for two dice just multiply 3.5 * 2 = 7

The answer is 7 because the expected value of rolling a single dice is 3.5 ((1/6 * 1)+(1/6*2) + (1/6*3) + (1/6*4) + (1/6*5) + (1/6*6)), so for two dice just multiply 3.5 * 2 = 7

The answer is 7 because the expected value of rolling a single dice is 3.5 ((1/6 * 1)+(1/6*2) + (1/6*3) + (1/6*4) + (1/6*5) + (1/6*6)), so for two dice just multiply 3.5 * 2 = 7

I think the reflip for the above isn't quite right. I think it's 2 + 15/16 * 1/2, since you have to account for what the reflipped coin may yield, not just the likelihood of having one tail.

I'd pick 14

for the dice question, the sum of the bottom half needs to equal the sum of the top half. The sum of 1:14 = 105, the sum of 15:20 = 105. You should choose to go first, depending on your risk aversion you should either choose 15 or 14

Odds of more than 2 heads: 5/16. Expected value is $2 with no re-flips. If allowed 1 re-flip, this adds $0.50 of EV for 15/16 of the situations (you don't added value if you got 4 heads the first time), so EV = $2 + 15/32 = $2.46875. For the dice question, EV is the same ($5.25) if you choose either 14 or 15, therefore, I would allow my opponent to choose first to give myself the opportunity to take advantage if he/she makes a poor choice. If he/she doesn't choose 14 or 15, I can choose the number that is 1 away from their choice on the side closest to 14.5 and I will have an EV greater than $5.25 due to their sub-optimal choice.

Expected value is not the same for range 1:14 it is 14/20*105=73.5 and for 15:20 it is 6/20*105=31,5. Sums are equal but you will be winning a lot more if you chose 14 and your opponent chooses 15.

Wrong calculation in above answer, Rance is right.