Before you choose one of the envelopes, one contains x dollars and the other contains 10x dollars. Once you make a random choice, there is 1/2 chance you picked one with x (case I), and 1/2 chance you picked one with 10x (case II). Expected value in the envelope you have is 1/2*x + 1/2*10x = 11/2*x. Then you have an opportunity to switch. In case I, you will now have 10x dollars, and in case II you will now have x dollars. The expected value for switching is thus 1/2*10x + 1/2*x = 11/2*x. Expected profit is the difference of switching value and initial value, which is 0. Switching has no impact on our profit, period. There is no paradox here, just bad analysis of the problem
It's a made up problem. As the brain processes sound relatively, it makes absolutely zero difference whether or not something is """perfectly in tune""" The difference between a perfect third, and an equal temperament third actually is none. The difference only exists when they're compared directly to each other through physical comparison, or by comparing your memory of the note, with what you're hearing in the moment.
Great video, thanks! I’ve always equated tuning systems to earths imperfect rotation and thus the need for leap-year. Calendars and tuning systems: The bane of humanity.
The expected amount of a random amount from the infinite set of all positive numbers is greater than any finite number. So any given number when the envelope is opened is always infinitely smaller than the amount in the other envelope. If the number is not simply drawn at random from all random numbers with equal probability, the other calculations change
This is fascinating. I'm an electronics engineer so I understand and work with harmonics and octaves all day long but have always wondered about musical tunings and why there are "missing" black notes. I shall now be able to write my masterwork with my discrete note theramin.
Hi sir - I have a question - so in a nutshell does it mean that the basis of any tuning in western music is to primarily achieve 2 goals - one is to capture perfect fifths of each note in the instrument and the 2nd goal is ability to transpose? I have no concept of music theory hence my question might be too naive... so if we have the keys of a piiano say tuned at any AP series, then transposition problem would be resolved but the "square root of 2"f formula addresses transposition as well as accomodates the perfect 5th (well not exactly perfect but almost) hence we chose that. Is that a correct understanding?
Hey there! I hope you're doing well. I wanted to get in contact with you regarding this research paper I wrote for tuning systems using this video as my foundation and just wanted to confirm some stuff with you! Thank you.
There is a aspect to the problem that really bothers me. Looking at the total contence of both enveloppes. It seems to be $11/2 + $110/4 + $1100/8 etcetera. So the average value would be infinite. That is bunkers!
Why do you assume the probability that the two envelops contain 10/100 is twice the probability that they contain 100/1000 (etc)? There is nothing in the problem statement to suggest this is so. Neither is there anything in the problem statement to say there are an infinite number of possible pairs. What if there are a largish number of x/10x pairs (and you do not know the number of pairs or the maximum payoff pair). Does the problem collapse? Alternatively, can you explain the apparent paradox under this alternative and equally valid set up? The set up you chose seems to have been selected to enable the solution you show. So your explanation only explains one highly constrained version of the problem.
You are right. The "paradox" exists only under certain specific versions of the story, such as the one with the specific amounts and probabilities I presented. I should have been clearer about this point in the video. The paradox disappears in some other versions, for example when there are only finitely many pairs of sums. Cheers!
As a guitar beginner, I strummed to play the harmonic above the 19th fret (B on the E string) to find it's just the (nearly) 1/3 length of the string, and B is the fifth note of E major. I proceeded to check the 1/2 length (the 12th fret) to find the same pitch name for 1/1 length, and guessed human brain just use the logarithm, whose base number is 2, to make the cycle. So the interval of half note is just geometrically dividing into 12 pieces, and what makes the fifth note so special is that 2^(19/12) ≈ 3. I've been looking for a music-theory book under advanced mathematics to get a further scope about harmony theory, which plays an important role in constructing chord on guitar. Thanks for affording the links below and your Manim programming is so fabulous for me to check the history of what I've known before.
This was such an amazing video. I only started taking interest in music a few years ago and never attended any lectures, so my knowledge of the more complex aspects of music theory is very limited. Because of this I often have problems understanding videos of this type, trying to explain the connection between math/physics and music. This was the first video I understood completely and it kinda blew my mind to be honest. Fantastic job. Thank you so much.
"Octave" indeed comes from "8", and that's because it's the eighth note in the basic scale we discussed (and is the most common scale in western music), see 12:45. Similarly, "fifth" because it's the fifth note in that scale. It has nothing to do with the equality 3 + 2 = 5. Cheers!
You open one envelope and find $40. That means the other has either $4 or $400. Sure, worth switching. But what if you happen to first open the envelope with $4 or $400 in it instead? Then for exactly the same reason it's not worth switching. Symmetry. You don't know which you have, so you don't know whether to switch or not. Symmetry.
The correct answer is to pick one and if there is an amount of money you would rather not lose in it (vs. an amount you care enough about not to lose) then keep it. This way you avoid the maximum feeling of loss in all situations. Details left as an exercise. ;)
If your envelope has an odd number of dollars in it, then you can deduce that you have the envelope with the small amount in it. ALWAYS switch if your envelope contains an odd number of dollars. The probably that your envelope has an odd number of $ is 1/2, if the amount of $ in the envelope is random. If a random amount of $ is a property of the problem, then you always have at least a 50% chance of getting the big amount.
Used often in choral music or chamber vocal music, its much nicer to tune chords individually to fit into just intonation, but the thing with vocal music (and fret-less string and brass too) is that there doesn't need to be a wolf interval because the music can be adjusted on the fly seamlessly. And especially for vocal music, many times the adjustment comes with little effort (except if someone has perfect pitch!)
