@@Kualinar This is intended to be obvious. If you are allowed to choose elements from {1, 2, ... , n} as you want, you are always choosing n and any other number you want, even 1 for example; in that case, you immediately reach a sum strictly bigger than n, so the expected value of m is 2 without any effort.
The exact wording from the thumbnail is : «How many element of {1,2....,n} until their sum is expected to be greater than n». The question NEVER imply any choice. It imply straight summation of elements. So, according to the original question, the answer it 2, as 1 + 2 = 3, and 3 > 2. In the video, this get changed to «chosen» without stating that any single value can be chosen more than once. This changes again later on.
This is very similar to "Sample numbers from the uniform distribution U(0,1) until the sum exceeds 1. How many samples are expected?" You can obtain e by using the Taylor series expansion and knowing that the volume of the n-dimensional unit simplex is 1/n!. Then you can transform this result to his problem by showing that in the limit they are nearly equivalent.
"Repeatedly roll a fair n-sided die until the sum of the rolls exceeds n. What is the expected number of rolls?" Would give a more precise formulation without complicating things
The problem is very poorly stated and is in fact different from what the thumbnail says. A different interpretation arises from that, for which the solution would be k = \ceil{2n/(n+1)}, which converges to 2 for large enough n. Incoherent thumbnails is a frequent problem in this channel.
If the entire problem is visible in the thumbnail, people can just try to solve it and leave without clicking on the video. This hurts the channel analytics and thus growth. It definitely can be irritating, but I see why they make the thumbnails a bit ambiguous. Just giving you the main idea of the problem without all the details. That way to see more you have to click on the video.
The set {1,2} has only two elements, so the largest number of elements anyone can choose from this set is 2, so the expected value you seek cannot be more than 2. Michael, your solution doesn’t apply to the stated problem. Thus us the trouble with starting problems non-rigorously. It’s easy to misinterpret it mathematically. A better way to state the problem you solved is the following: “What is the expected length of a sequence of elements from the set {1, …, n} so that the sum of the sequence is a than n?” I realize that the difference may be thought to be subtle, ands accordingly, I’ll likely be called a pedant, but some students go watch might take a class from someone as pedantic as an I and lose points on an exam because they misinterpret questions this way. To drive the point home, and at the risk of appearing even more pedantic, let me restate the original problem to have the same meaning as the original problem but in a more rigorous manner: “What is the expected cardinality of a subset of the set {1, …, n} whose sum is greater than n?” For this problem, the answer, as I mentioned above, cannot be larger than n.
Yes, the original statement of the problem fails to clarify whether the same element can be used more than once in the sum. Obviously from the way Michael gave the example and the way he solved the problem he intended to allow reuse of the elements. An interesting follow-up question would be the similar problem but without allowing any element to be used more than once (i.e., "without replacement" in the language of intro probability problems). I think the limit would still be e, but the expression for the expected value would differ.
This is definitely a non-formal (i.e. non-mathematical) question, resting on the interpretation of words that are not properly operationalized. It gets turned into a formal question in the video, but the choice of interpretation for the word "chosen" (that you can choose the same number more than once) does not match the "choose" operation typically used in combinatorics, which is the field this question rests in, so they are relying on people going to informal language and taking a different "common" meaning. These types of tests should always be called out, as their lack of a true mathematical / formal approach without fudging is one of the reasons people struggle with mathematics, and they aren't testing what they think they are.
I assume Michael tried to word it so that it is easy to understand, but he made it ambiguous in the process. Most likely the original statement was rigorous enough, but it was wordy
Wouldn't it be better to phrase the problem as: "What is the expected value of the *minimal* number of elements from {1,2,...,n} that *can* be chosen *with* repetitions (with the probability of choosing any element equals to the probability of choosing any other element) so that their sum is greater than n?"
Actually, in the context of applied probability it is a well known fact that for ANY random variable X whose support is a subset of the natural numbers, E[X] = sum over k=1 to infinity of P(X≥k). The proof follows from writing P(X≥k) as a sum from j=k to infinity of P(X=j) and swapping the sums. Now, in this problem the support is finite, so you can truncate the sum at n, and you get your result.
Looked at the thumbnail and thought, "that's probably going to spit out an e." That's a rewarding feeling! The criticism of the thumbnail is valid, though the infinite result would be the same. Interesting problem!
It took me a minute to understand the premise. What helped me understand was reinterpreting the question as “what is the expected number of cards needed to pull in order to bust the sum of n? Assume you’re pulling from a deck of cards ranging from 1 to n.” It’s like blackjack but more generalized. I don’t understand much of what’s being done in this video, so my personal approach will be very amateurish and clunky, but in the spirit of Matt Parker, I’m giving it a go. Edit: I’ve noticed that partitions help a lot when trying conceive this problem under any particular n. You find the number of ways n can be partitioned, then find how many numbers every partition is made from, and that is how many cards you would have to pull (or more aptly, how many dice would need to be rolled). There’s more to it, but it’s cool to notice this
I wondered what if the sum needed to be larger OR EQUAL to n and I reached (1+1/n)^(n-1). Is it correct? Because interestingly, if so, it is e as well when it goes to infinity
Yesterday i saw Dr Sean's video about 5 different complexities of e,and when i saw this i was fascinated e appeared in seemingly unexpected problem.Great to see in depth explanation, mind blowing stuff
Wanted to add a snarky comment during calculations "waiting for the sudden emergence of e or pi" and in the last seconds of the material - here we are! You snarked me XD
That is the probabilty you need to achieve a sum greater than 2 if you choose 1 element OR 2 elements OR 3 elements. The expected value E[X] is the mean of the number of elements you need to choose, and it is a wheighted mean (the wheights being their respective probabilities).
I really wanted to know a rigourous definition of a tangent, my teachers say its a line which passes through a single point on a curve without intersecting then I ask them how to define intersection rigourously and they have no answer, some people give me the calculus definition but I already know it and its not a satisfactory answer to my question
Assume there is a function f(x). A line L(x), which is a linear function with constant slope, is tangent to f(x) at some point x_0 if L(x_0)=f(x_0) and there is some neighborhood (a,b) about x_0 such that for y in (a,b) where y is not x_0 it is the case that L(y)=f(y). There also has to be a condition on f(x_0) relating to its derivative, but I think this helps establish a baseline on the tangent line.
Perhaps if you know multivariable calculus you can look at tangent vectors/tangent planes/curvature to get something that generalizes to higher dimensions?
I don't think that definition is correct. The x-axis is tangent to the curve y=x³. Is wikipedia on "tangent" not satisfactory? "More precisely, a straight line is tangent to the curve y = f(x) at a point x = c if the line passes through the point (c, f(c)) on the curve and has slope f'(c), where f' is the derivative of f."
I would say, that a tangent line to a curve is a line such that they share only one point and all other points of this curve lie to one side of a tangent
@@LookAsLukas But that's not true, a curve is allowed to cross its tangent line, or you could even have a curve in 3 or more dimensions where "one side of a line" is meaningless
The idea is to create a sequence out of the numbers 1 to N. Like 4,5,7,4,7,1,... If the sum should be bigger than n, you need at most n+1 elements (1+1+1...+1)
Hi Michael Penn! For a nice little physics problem I have derived a very difficult differential equation: x' '(t)=(g*a/l^2*b)*sqrt(l^2*x^2-x^4). I have tried to solve it using wolframalpha but it didn't solve it so if you can solve it it would be very nice. Best regards!
