Common Mistakes in Problem Set SIX
Problem 1 (Tom) [4 points]
Two points were assigned to each part, with one point for the correct answer (true/false) and one point for the proof. Few people had trouble with this problem. One thing to note is that you don't know that sets form an algebra over union and intersection -- that is, you can't claim without proof that A intersect (B union C) = (A intersect B) union (A intersect C). This is easy to show, either by considering an element of each side in turn, or using set-builder notation, but you need to show it.
Also, while we did accept Venn diagrams as proofs for this problem, there exist problems where it is effectively impossible to consider all possible arrangements of the sets (e.g. with 4 sets you might need to consider as many as 65536 arrangements). In general, a mathematical proof is safer than one involving Venn diagrams (though the diagrams remain invaluable tools for thinking about the sets).
Problem 2 (John) [3 points]
I think everyone pretty much knew the answer (I recall only one person getting it wrong) but a number of people had problems with the proof.
The most common error was the thought that Pascal's triangle showed that it was the middle without relating Pascal's triangle to the problem, or doing induction on it.
Problem 3 (Yogi) [4 points]
Most of the students got this problem right, and those who did not got almost no credit. A very few students were there who made some small mistakes and hence got only a partial credit.
One mistake that many students made (and I did not take off points for that) is that they did not explicitly mention that {n \choose -1} is zero or {n \choose n+1} is zero. The best thing was to drop the term corresponding to k=0 in the summation (as it had multiplication with k), otherwise, the cases of choosing negative number of items should have been handled correctly.
Some students did the proof by
induction, and
were not comfortable with the step 2 where we substitute k-1=j. If you went
only will step 1 above, you collected 2 points (out of 4 possible).
Problem 4 (Tom) [3 points]
The intention was that one point be given for the answer, one for an example when the maximum is attained, and one point for the proof of maximality. Due to a misunderstanding, the actual rubric was one point for the answer and two points for the proof. This was applied across the board, with the result that almost all grades were 1 point harsher than intended. I apologize for this, but the grades are all consistent; as a result, please do not ask for a regrade just because you did not get credit for including an example. You should still submit a regrade if you feel there is any other problem with the grade.
The biggest problem that people had was in proving their answer of n correct. The following argument does not suffice: "Let A be the collection of all singleton sets; then |A| = n. And if we create A' by adding any other set to A, there will be two sets which overlap, disqualifying A'. Thus A is optimal." The issue here is the assumption that A is a subset of A', which is not necessarily true. A proof needs to consider all possibilities for A', not just those A' which resemble your A. The heuristic argument "any set which has more than one element 'eliminates' more elements than it needs to" is a good way to think about the problem, but should be made more rigorous. A pigeonhole argument could be made to work if you were careful about what you were claiming.
Problem 5 (Kevin) [4 points]
1 point for getting P(n,k) correct.
1 point for getting F(n,k) correct.
1 point for the explanation of P and F.
1 point for giving the correct answer for F(n,k) when k=1 and when k=2.
The most common mistake on this problem was either explicitly or implicitly assuming that the order of the purchases doesn't matter. The problem was dealing with "sequences" and "histories", so order did matter. Since it wasn't outright stated in the problem that order matters, only one point was deducted for making this incorrect assumption if the rest of the analysis was correct under that assumption (P(n,k) = nCr(n + k - 1, k) and F(n,k) = n^2).
Many people didn't cover the case when k=1 or when k=2. The general answer for F(n,k) is actually incorrect for these two cases, so they needed to be discussed separately.
Some people forgot that a highly focused history also includes cases where all the genres are the same. This means you need to add (n) to your term (n(n-1)k). Also, some people forgot to multiply (n(n-1)) by k to account for the odd-man-out genre being placed in any position of the sequence.
Problem 6 (Joel) [4 points]
There are two mistakes that were very common. It was neccesary to show that 2^{n-1} was both an upper bound for the size of Q, and that it was attainable. In other words, you need to show that you can get 2^{n-1}, and that you can't do better.
Problem 7 (Yogi) [Optional 0/1]
This was a difficult problem compared to the last problem sets' optional problem. There were three parts to the problem, and computing s(n,k) was the most difficult. Unfortunately, due to the grading scheme on optional question, if you attempted two easy parts but did not find the expression for s(n,k), you did not get the credit. Some common pitfalls were the following:
Some students said that they found the particular formula with ``brute force'' and wrote that as their answer. You were needed to supply the proof/argument how you got the particular expression for s(n,k), failing which you did not receive any credit (even if your expression happened to be correct).
Some student made the mistake that they said that we can select n-2k position in just n-2k+1 ways because they could either be all one, or all zero or 1's followed by zeros. This is true if you are not allowed to break up this set of n-2k positions. But these n-2k position could have been divided in k+1 continuous chunks, and hence the above argument works for each of those chunks, but not for all n-2k positions. We have to take care of number of ways in which n-2k position can be partitioned in k+1 parts.
Please see the solutions (available on CMS) for further details.
Common Mistakes in Problem Set SEVEN
Problem 1 (Tom) [3]
Almost everyone got this right. A few people used the division rule, but instead of dividing by (3!)(3!)(2!)(2!)=144, divided by (3)(3)(2)(2)=36. Also, some people went through the problem letter by letter, putting in the 3 E's, then the 3 S's, etc. This gave the right answer (and most people did it correctly), but it requires a lot more work, and it's easier to make a mistake.
Problem 2 (Yogi) [3]
Students did very well on this problem (so there are not many "common" mistakes as such). Some of them are:
The off-by one problem was there. Some students forgot to subtract one from blocked configuration (one state in which all processors are done is not blocked).
A couple of students thought that the ordering of the processors does not matter (they took processors as indistinguishable). It was not very clear from the question whether ordering matters, but in case of doubt, you should make it clear in office hours, or atleast state the assumption in the answer. Anyway, the problem was more difficult if you do not take ordering into account, and you did get credit if you thought the processors as indistinguishable.
Some students gave the answers for particular values of n, say for n=3, n=4 etc. The question asked the expression in terms of n (and not for any particular value of n).
Problem 3 (Zach) [2+2]
This problem was worth four points, two for each part. To earn full credit, you needed to have given a correct answer, and a valid explanation. The explanation was worth 1 point, and the correct answer was worth 1 point.
For number 3, the biggest problem I saw was lack of explanation in derivation of the answer. You needed to have given a clear reason why the number of multiplexed streams is the same as the number of ways of rearranging aa...abbb....b, etc. Many people just used this, but did not say why the two numbers are the same.
Problem 4 (Hari, John) [5]
Check the final solution you give closely. It will help you show that is satisfies the properties, and it will prevent a silly mistake by giving a solution that can easily be broken. If you find that your solution can be broken, you likely did the whole problem wrong.
Also, be rigorous with a proof. Don't give an example of how the scheme fails and say that the example proves it. DON'T cover huge swaths of your proof with things like "obviously" and "from this it clearly follows" unless it REALLY clearly follows. If you have a doubt just put in a line explaining why it is so clear.
Problem 5 (Tom) [3]
Most people received full credit for this problem; it was very easy, requiring one short application of the inclusion-exclusion rule.
Relatively few people did it correctly, though; many people had problems with the definition of an event and even a sample space.
Please see the solutions, as less than 20% of the class correctly identified the event. The event is not the sentence "5 | k or 7 | k".
The event is a set, and it is a subset of the sample space. Some people gave heuristic arguments, which was all right, but then neglected to mention the sample space or event at all. We were very lenient regarding probability and proofs on this first homework, but until you have the concept of independent events, truly correct heuristic proofs are almost impossible.
Problem 6 (Kevin) [2+2+2]
Nearly everyone got these right. The only mistakes were
leaving out a description of the sample space (which was actually all possible results of throwing k dice, but we accepted anything reasonable) and
forgetting to filter out all events which didn't meet the criterion, i.e., that the sum of the first two dice is 7 or that the sum of the first three dice is 8.
Problem 7 (Joel) [0/1]
The most common mistake was not using the proper definition of the Quorum system from the previous assignment.
Only very few students got this problem right. Given that the problem was comparatively easier then the optional problem of other problem sets, it seems that students have stopped attempting the optional problem. The students should atleast try to attempt the optional problem...
Common Mistakes in Problem Set EIGHT
Problem 1 [3] Joel
The only mistake on problem one was to fail to mention that you were calculating the expected number of votes for democrats, not the actual number.
Problem 2 [4] John
On this problem most people knew how to compute the answer to the problem, and the rules for the conditional probability were pretty well understood.
However, a great many people mistook the sample space to be emails from hotmail rather than all email to Cornell.
Problem 3 [2] Tom
Every single person did this problem correctly. A handful of people made arithmetic mistakes, but were still doing the right thing. It can be a good idea to check your answers to make sure they make sense -- if you get something that isn't anywhere near what you expected, try going through your calculations again.
Problem 4 [7=4+3] Hari
For the most part, people seemed to get this. However, there were a few problems.
People said that the probability of one machine getting into S is 1/2^(d+1) so the expected number of machines getting in is n/2^(d+1). While this is true, you have to make some reference to the linearity of expectation. If this statement was not properly justified they were given the comment "Why?"
The only real problem here is that people didn't follow directions and many forgot to give the Expected number of machines in S given the value of p they derived. For this they
received the comment "E(X)?" Also, points were not taken off for people who did the calculus wrong.
Problem 5 [8=1+2+2+2+1] Kevin (black fonts), Zach (blue fonts)
Most people got this part right.
No comments.
The most common error was forgetting to multiply 1/k * (1 - 1/k)^(k-1) by k; this is necessary because each computer can receive any of the k different jobs. Without this extra coefficient, you found the probability that a given machine receives exactly some pre-specified job.
Many people started out correctly and yielded (1/k)(1-1/k)^(k-1). But, this is the probability that a given job i, goes to machine j, and the other k-1 jobs
don't go to machine j. The problem with this is that there are k choices for i, so we must multiply through by k. This yields (1-1/k)^(k-1).
A lot of students tried to take the k-th power of their answer for part (b) as their answer for part (c). This is not right because the event that some machine gets exactly one job is not independent of the event that some other machine gets exactly one job. The best way to approach this problem is to notice that there are k^k ways of assigning jobs to processors, and k! of these ways result in one job per processor. So the correct answer is k!/k^k.
A lot of students tried to prove that it converges to 0 by noticing that the numerator is always smaller than the denominator. This does not prove convergence to 0, e.g. consider (k-1)/k. This converges to 1. You could have shown that k!/k^k is bound by 1/k, which goes to zero, or you could have used the ratio test.
For this, many people took the answer from b) and raised it to the kth power, using the rationale that the probability of a machine getting one job is (1-1/k)^(k-1), so the probability that machine 1 gets one job and machine 2 gets one job,..., and the probability that machine k gets one job is just ((1-1/k)^(k-1))^k. This is not correct, as the probability that machine a gets one job is not independent of the probability of machine b getting one job (because if machine a gets a job, there is one less job for b to be able to get so the probability is no longer (1-1/k)^(k-1)). What we do instead is to say: there are k^k ways to distribute the k jobs (each job has a choice of k machines), but only k! ways with each machine getting one job (job 1 goes to one of k machines, job 2 goes to one of k-1 jobs, etc.), so the probability is k!/k^k.
Now, in proving that the limit of k!/k^k = 0 as k -> infinity can be done in several ways, but it does not suffice to say that the denominator is growing faster than the numerator as this is not rigorous and does not actually prove that the answer is 0. It is easier to say:
k!/k^k = 1/k * 2/k * ... * (k-1)/k * 1 <= 1/k as each of the other terms in the product are <= 1. Therefore, limit k!/k^k <= limit 1/k = 0, and we get the result as k!/k^k is nonnegative for each k.
Also, some people tried using the ratio test which does work, but the ratio test is used only for infinite sums, so it only works in this case by careful analysis. Using the ration implies that the sum from 1 to infinity of n!/n^n converges. And as it converges, its terms must go to 0, so limit of n!/n^n -> 0. Try to stick to more conventional ways of proving limits like comparison and algebra.
By the linearity of expectation, we can just multiply our answer from part (a) by k. Otherwise, this quantity is not difficult to calculate directly; most people got it right.
The biggest mistake I saw was that people misread the question and tried taking the limit of N(k), while the question asked for N(k)/k which converges to 1/e.
Everyone should have realized that R(k) = N(k). Without this fact, it is very hard to find the value of R(k) directly. Few people did so correctly.
The trick to this problem, and the only sensible way to do it by avoiding horrible calculations, is to notice that R(k) = N(k), because the number of rejected jobs equals the number of machines with 0 jobs, since if a job goes to a machine which has a job already, then some other machine cannot receive that job, and as the number of jobs equals the number of machines we get that one machine must not receive anything. Extending this argument to all the jobs gives the result.
Problem 6 [0/1 Optional] Yogi
Not many students attempted the problem but the statistics were better than the last time (for optional problem). The common mistakes were the following:
Some students just mentioned their strategy without any justification or proof. They mentioned a sequence of steps that the seller should take (without any justification), and sometimes the steps were not even exhaustive (they did not cover all the cases). In such cases, no points were awarded.
Some other students tried analyzing the strategy, went quite far inside the explanation and proof, got some complicated expressions and then just said that the expression they got is greater than 4 (again without justification). Please note that you should provide justification to all the steps you write unless the step is completely obvious/trivial.
There was one common mistake about the independence of the events. If the highest bid is in the second half of the sequence, then the probability of second highest bid being in the first half is not 1/2 but a bit more then that which is n/(2(n-1))>1/2. But we did not take of (sole) point for this mistake. But please be careful when you multiply two probabilities, the events have to be independent to multiply the corresponding probabilities.
Just a final note which is not specific to this problem but applies in general. Students are (sometimes) doing addition/multiplication of probabilities in an ad-hoc manner, without justifying the operation. Please convince yourself before the operation you do. For addition of probabilities, you need the events to be disjoint and for multiplication, the events must be independent. It might not be very crucial (to understand all this) now, but once the problems become complicated enough, your arguments needs to be sound.
Problem 1 (3) Eva/Hari
Despite the close similarity with Question 5 from Problem Set 6, many people did not think of handling separately the cases when fewer then 3 genres are involved in a sequence. This lead to overcounting the possible solutions. Also many people did not handle short sequences separately: when k=3 or smaller, all sequences are almost focused, and the formula derived for higher values overcounts this. In fact, there is even a case where k=4 has to be handles separately. Very few people noticed this fact, and we did not subtract credit for this.
Problem 2 (3) Hari
People who approaches this with induction tended to get it right. Some people didn't use Pascal's theorem in the inductive step and ended up doing a good amount of needless algebra, but points were not taken off for this.
A few people tried some combinatorial arguments. While this can work (see the solutions) only one person did this correctly.
A good number of people tried the straight algebraic argument. Often times people would manipulate the left side of the equation and after a couple of steps, would magically deduce that the expression they had was equal to the right side. Points were deducted for jumps in logic such as this.
Problem 3 (3) Tom
Most people who made mistakes on this problem gave one of the following answers: "100^3"; or "C(100,3)"; or "C(100,3) + C(100,2) + C(100,1)". The correct answer was "C(102,3)" and can also be written as "C(100,3) + 2*C(100,2) + C(100,1)", "100 + 100*99 + 100*99*98/6", or "171,700". There were other correct ways to write the answer.
The answer "100^3" comes from not realizing that order doesn't matter; the answer "C(100,3)" comes from not realizing that items could be repeated; the answer "C(100,3) + C(100,2) + C(100,1)" comes from not realizing that "banana,banana,apple" is different from "banana,apple,apple". These responses all warranted partial credit.
Problem 4 (8) Tom
It MIGHT suffice to show that the distribution is symmetric about n+1, but we don't have the tools in this course for that (and it's not necessary).
Problem 5 (11) Kevin + Zach
I really don't have any informative comments for my problem, other than to read the solutions.
Problem 6 (12=4+4+4) Joel + Yogi.
The most common mistake was to forget n! or 2^n in the formula for number of pairing (correct formula is (2n)!/(n!*2^n)). This was mainly because some of the students did not consider that the ordering of all the pairs does not matter (it does not matter whether I call the pair of (A,B) first pair of i th pair).
Some of the students misunderstood the question to find the number of possible pairs that we can select from 2n students... the answer to that is clearly {2n \choose 2} and the problem became very easy after that. The maximum credit you got (in part a. and b. combined) after you made this mistake was 3/8.
Some students made the mistake in third part by summing the expectations for only n students. The expectations were to be added for all 2n students. You lost one point for this mistake.