Baruch MFE Spring 2012 Baruch MFE Facebook Contest

[dstefan]

Spring 2012 Baruch MFE Facebook Contest​

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Monday, February 6 - Saturday, February 11​

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The second Baruch MFE Facebook Contest will take place over six days beginning on Monday, February 6. One brainteaser question will be posted every day at 12noon New York time (EST) on the Baruch MFE Facebook page, beginning on Monday 02/06 through Saturday 02/11. The contest is open to people all over the world.

The questions will be similar to phone interview type questions, requiring short answers. Sample question:

A given organism can either die, do nothing, split into two organisms, or split into three organisms with equal probabilities. If you start off with 1 organism, what is the probability that the organisms eventually die off?

Answers will be posted as Comments on the Facebook Page.

Prizes:
1st Prize: Signed copy of A Primer for the Mathematics of Financial Engineering, Second Edition and its Solutions Manual, by Dan Stefanica, the Director of the Baruch MFE Program, and an invitation to visit the Baruch MFE Program, meet with faculty and students, and sit in the class of your choice (or phone conversations with two Baruch MFE faculty members, if not located in the NY area)
2nd Prize: Signed copy of "A Primer for the Mathematics of Financial Engineering, Second Edition" and its "Solutions Manual" and a meeting with the author (or phone conversation, if not located in the NY area)
3rd Prize: Signed copy of "A Primer for the Mathematics of Financial Engineering, Second Edition"

Scoring:
2 points - every correct answer submitted in the first 5 minutes after the question is posted
1 point - every correct answer submitted in the first 30 minutes after the question is posted,
but later than the first 5 minutes
3 points - the best answer submitted within one hour of the question being posted

Good luck, and looking forward to it!

 

[darth]

Can we have the question posted on QN as well at the same time for people who do not have a facebook profile.
Moreover facebook is locked in many offices , so it will make it difficult to post a quick answer.

Also if the best answer is submitted within 5 mins... Will it get 5 points?

 

[dstefan]

Can we have the question posted on QN as well at the same time for people who do not have a facebook profile.
Moreover facebook is locked in many offices , so it will make it difficult to post a quick answer.

The questions will only be posted on Facebook, which is where they should be answered as well.

Also if the best answer is submitted within 5 mins... Will it get 5 points?

It is 3 points.

And yes, it will, but within the first 5min it would make more sense to only provide a number which is the solution. The elaborate version of the answer should come within an hour and be good enough (albeit brief) to warrant the 3 points.

 

[babinu]

The questions will only be posted on Facebook, which is where they should be answered as well.



It is 3 points.

And yes, it will, but within the first 5min it would make more sense to only provide a number which is the solution. The elaborate version of the answer should come within an hour and be good enough (albeit brief) to warrant the 3 points.

Cool great, is it necessary to register for this event ? or is having a facebook account good enough ?

Thanks
Babinu

 

[dstefan]

No need to register. Having a Facebook account will do. Good luck!

 

[dstefan]

As preparation for the Facebook contest starting next week, you could try solving the sample question below:

A given organism can either die, do nothing, split into two organisms, or split into three organisms with equal probabilities. If you start off with 1 organism, what is the probability that the organisms eventually die off?

None of the answers posted so far on Facebook at http://www.facebook.com/BaruchMFE are correct.

All questions will be similar to phone interview type questions, requiring short answers.

 

[dstefan]

Sample question:

A bug starts at a vertex A of triangle ABC. It then moves to one of its two adjacent vertices. How many paths of length 8 end back at vertex A?

Post answers at
http://www.facebook.com/BaruchMFE

 

[dstefan]

Starting in 2 hours!!!

The second Baruch MFE Facebook Contest will take place over six days beginning on Monday, February 6. One brainteaser question will be posted every day at 12noon New York time (EST) on the Baruch MFE Facebook page, beginning on Monday 02/06 through Saturday 02/11. The contest is open to people all over the world.


Scoring:
2 points - every correct answer submitted in the first 5 minutes after the question is posted
1 point - every correct answer submitted in the first 30 minutes after the question is posted,
but later than the first 5 minutes
3 points - the best answer submitted within one hour of the question being posted


Prizes:
1st Prize: Signed copy of A Primer for the Mathematics of Financial Engineering, Second Edition and its Solutions Manual, by Dan Stefanica, the Director of the Baruch MFE Program, and an invitation to visit the Baruch MFE Program, meet with faculty and students, and sit in the class of your choice (or phone conversations with two Baruch MFE faculty members, if not located in the NY area)
2nd Prize: Signed copy of "A Primer for the Mathematics of Financial Engineering, Second Edition" and its "Solutions Manual" and a meeting with the author (or phone conversation, if not located in the NY area)
3rd Prize: Signed copy of "A Primer for the Mathematics of Financial Engineering, Second Edition"

 

[dstefan]

Question 1:

A stick of length 1 drops and breaks at a random point distributed uniformly across the length. What is the expected length of the smaller part?

 

[dstefan]

Three different answers warranted 3 points:

Juan Pablo Alonso
The smallest part will always be less than or equal to 0.5. The probability distribution can be defined as two separate functions:
Case A, If the stick breaks between 0 and 0.5 (considering the left of the stick to be 0), the small part will be on the left side with expected value E(A)=(0.5-0) /2, using the uniform distribution expectation formula.
Case B: If it breaks between 0.5 and 1, the small part will be on the right side, with E(B)=(1-0.5)/2, also a uniform distribution.
Since both cases have an equal possibility of happening, the expected value of the stick length will be P(A)E(A)+P(B)E(B)=0.5*0.25+0.5*0.25=0.25

Li Zhou
integral (min(x, (1-x)) dx from 0 to 1 -> by symmetry, for all x> 1/2, we use 1-x (since x > 1-x when x > 1/2). Therefore, integral (x dx from 0 to 1) + integral (1-x from 1/2 to 1) = 1/8 + 1/8 = 1/4

Francesc Rosell Sanz
‎1/4 since min(x,1-x) has a triangular distribution with support 0 to 1 and peak 0.5...the area of min(x,1-x) is 0.25...which equals the expected value of min(x,1-x) given that x follows a uniform distribution...

 

[dstefan]

Standings after the first question:

Juan Pablo Alonso 3
Li Zhou 3
Francesc Rosell Sanz 3
Jin Kong 2
Anthony Chi Wa Iao 2
Ayush Priyank Pandey 2
Raghuram Reddy 2
Lily Xiang 2
Dieter Dijkstra 2
Luis Toro 2
Yongyi Ye 2
Melvin Sim 2
Long Zhao 2
Nan Li 2
Adrian Garza 1
Gautam Punukollu 1
Prashant Kumar 1
Nanda Kishore 1
Tuan Le 1
Babinu Uthup 0
Ken Kruger 0
Teye Brown 0
Alireza Kashef 0
Leo Yang 0
Liyun Dong 0
Shubhankit Mohan 0

 

[dstefan]

Question 2:

Consider a deck of 52 cards. I pick one card first, then you pick one. What is the probability that my card is larger than yours?

 

[dstefan]

Three point answer (also voted by the audience ):

Dieter Dijkstra
After I pick one, player 2 has 3/51 (ie 1/17) chance of picking the same rank card, and it's clearly symmetric who has the largest card. So that leaves each player 8/17 to have the largest card.

 

[dstefan]

Questions 2 scores:
Dieter Dijkstra 3
Jin Kong 2
Lily Xiang 2
Luis Toro 2
Nan Li 2
Babinu Uthup 2
Amit Gupta 2
Li Zhou 1
Yongyi Ye 1
Long Zhao 1
Gautam Punukollu 1
Prashant Kumar 1
Nanda Kishore 1
Tuan Le 1
Yibing Tang 1
Xman Wang 1
Trent Kilpatrick 1
Zinoviy Mazo 1
Juan Pablo Alonso 0
Ayush Priyank Pandey 0
Raghuram Reddy 0
Alireza Kashef 0

 

[dstefan]

Overall:

Dieter Dijkstra 5
Jin Kong 4
Lily Xiang 4
Luis Toro 4
Nan Li 4
Li Zhou 4
Francesc Rosell Sanz 3
Yongyi Ye 3
Long Zhao 3
Juan Pablo Alonso 3
Anthony Chi Wa Iao 2
Melvin Sim 2
Babinu Uthup 2
Amit Gupta 2
Gautam Punukollu 2
Prashant Kumar 2
Nanda Kishore 2
Tuan Le 2
Ayush Priyank Pandey 2
Raghuram Reddy 2
Adrian Garza 1
Yibing Tang 1
Xman Wang 1
Trent Kilpatrick 1
Zinoviy Mazo 1

 

[dstefan]

Question 3:

Find the limit as x goes to 0 of
1/sin(x^2) - 1/(sin?(x))^2

 

[dstefan]

Three point answers:

Babinu Uthup
Use the mclauren series expansion for cosec(x) = 1/x + (1/6)x + (7/360)x^3 + ..... hence cosec(x^2) = 1/x^2 + (1/6)x^2 + (7/360)x^6 + .... cosec^2(x) = 1/x^2 + 1/36(x^2) + 2*1/x*(1/6)*x + terms involving higher powers of x. Subsctraction both gives the required expression as -1/3 + 1/36(x^2 + terms involving other powers of x. Hence the limit gives the answer as -1/3

Luis Toro
Aplying l'hopital:
Limx-0 (1/senx2 -1/sen2x)
=lim (sen2x-senx2)/sen2xsenx2
= lim (sen2x-senx2)/x4
=lim(sen2x-2xcos x2)/4x3
=lim(2cos2x+4 x2cos x2-2cosx2)/12x2
=lim(-4sen2x+8xcos x2+8 x3 x3cos x2+4xsen x2)/24x
=lim(-8cos2x+8cos x2-16 x3sen x2…..)/24=-1/3

 

[dstefan]

Question 3 scores:
Luis Toro 3
Babinu Uthup 3
Lily Xiang 2
Li Zhou 2
Yongyi Ye 2
Anthony Chi Wa Iao 2
Zach Lubarsky 2
Jin Kong 1
Nan Li 1
Juan Pablo Alonso 1
Prashant Kumar 1
Tuan Le 1
Trent Kilpatrick 1
Liyun Dong 1
Alireza Kashef 1
Dieter Dijkstra 0
Long Zhao 0
Amit Gupta 0
Ayush Priyank Pandey 0
Raghuram Reddy 0
Shubhankit Mohan 0

 

[dstefan]

Overall Standings:
Luis Toro 7
Lily Xiang 6
Li Zhou 6
Dieter Dijkstra 5
Jin Kong 5
Nan Li 5
Yongyi Ye 5
Babinu Uthup 5
Juan Pablo Alonso 4
Anthony Chi Wa Iao 4
Francesc Rosell Sanz 3
Long Zhao 3
Prashant Kumar 3
Tuan Le 3
Melvin Sim 2
Amit Gupta 2
Gautam Punukollu 2
Nanda Kishore 2
Ayush Priyank Pandey 2
Raghuram Reddy 2
Trent Kilpatrick 2
Zach Lubarsky 2

 

[dstefan]

Question 4:

An ant is in the corner of a cube. The ant moves randomly from one corner to another along the edges of the cube with equal probability in all the three directions. What is the expected number of steps for the ant to reach the opposite corner?