[ASMR] The Two Envelope Paradox 

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This is all I needed after a long day at Uni. Your voice is calming and clear. I’m going to sleep well tonight

Today's lesson is: Don't be greedy 😅

The problem with the compounding reward when repeatedly switching is the implicit assumption that there actually is an infinite series of envelopes (Y)_n, where the n’th envelope Y_n has the distribution P(Y_n = 2 * Y_{n-1}) = P(Y_n = 0.5 * Y_{n-1}) = 0.5, which gives the expected value E(Y_n) = 1.25*Y_{n-1} In that sense, it’s basically a discrete model for a volatile stock, or repeatedly betting half of your current chips on red in roulette (except that is slightly skewed towards you losing half).

Considering the amount of ASMR videos and math videos I watch, I'm surprised youtube took this long to recommend this channel.

In maths there’s something we call independent events and it is when the 1st event doesn’t effect the 2nd one which is in this case choosing A(1st) and choosing B(2nd) because choosing the 1st envelope will not have any effect on the probability of you winning the prize as well as the 2nd

need more of these type of problems videos!

i love your videos about these kinda problems!! super entertaining and interesting

This seems similar to the Monty Hall problem, with the doors and goats and cars and stuff.

Saying that envelopes contain "some amount of money" is saying that the amount is a random variable. That means there is a random variable we can call A, with a set of possible values {A1, A2, ..., AN}, and a corresponding set of probabilities for each {P1, P2, ..., PN} such that P1+P2+...+PN=1. This set is called the probability distribution for the random variable A. Then, saying that, as far as we know, the two envelopes are equivalent (well, you didn't say it, but you implied it) means that the random variable B has the same set of possible values and the same probability distribution. But then saying that the realized value of B is either half, or twice, that of A means these two random variables are not independent. And that means you need to use a joint probability distribution to perform the expectation calculation at 6:30. It is not EXP(B|A=$10) = $5*1/2 + $20*1/2, it is EXP(B|A=$10)=$5*Pr(B=$5|A=$10)+$20*Pr(B=$20|A=$10). And if you don't know that A=$10, you need the expectation over all possible values of A. And finding another way to calculate the expected value when switching does not prove that either is correct or incorrect. The one at 6:30 is wrong, because you need to know the joint probability distribution to use it. The later one should be right, because you don't need it. But, in theory at least, it doesn't have to be. In fact, there are distributions where there is an expected gain when you switch (the first time, not the second). But then the expected value of A is infinite, so it is unrealistic. In the real world, the fact that there must be an upper limit to AN means the expectation takes a huge hit when X=AN, since Pr(B=2*AN|A=AN)=0.

more asmr like this !!!! first video i seen on your channel and i love it!

Your whispering was great in this one!

Me partying at 12 at night when I have to be awake In 6 hours because I guessed B had more money

Relaxing. I first thought it was wrong all the way but you said it in the end so 👍

i love your asmr math channel. i also like your British RU-vid name and accent. the British tends to call mathematics maths in abbreviated form.

So awesome. I’m not that great at math but i love these interesting concepts.

This is great. Nice to meet you

Doesn’t matter for me. 10 bucks are 10 bucks I was happy that I even get something 😂

Not sure if this is a real thing - like an Actual paradox. But I love ASMR that I feel like I’m learning something 😂

Great video! I'd think another fascinating probability problem to cover would be the sleeping beauty problem if you can.

It is the stuff that I really like 😊

Hey Dido. Big fan of your channel and was wondering if you could go over trigonometry because I’m having difficulty with it in class. I live in Canada so I don’t know if trigonometry is taught differently in Scotland but I think it would help overall if you went through it.

Dido could you please make a video teaching fractions and %?

GOOD CHANNEL!!! idk if you've seen the movie kill bill but you look like a handsome version of bills brother. ur also rlly good at asmr and ya its just lovely

I love how the shape of the mic ( I think) looks like a potted tree

Can you do a video explaining the “Birthday paradox”

More paradox asmr pls

Expected value only makes sense when calculated for the two possible 𝘴𝘸𝘪𝘵𝘤𝘩𝘦𝘴. One switch is from $10 to $20. Call that x₁ and its probability p₁ The other is from $20 to $10. Call that x₂ and its probability p₂ . EV = (x₁ x p₁) + (x₂ x p₂) If x₁ = $10 then x₂ = $ -10 p₁ = 1/2, p₂ = 1/2. So we have EV = ($10 x 1/2) + ($-10 x 1/2) = $5 + $-5 = $0.

Hi dido i couldnt really understand the final anwser to the question. Basically what i understood is that if there are no values given, we should switch, but if they give us concrete amounts we shouldnt? Like if they tell us theres 10 dollars in on and 20 in the other thats when it doesnt matter if we switch or not🤔 anyways i really enjoyed the vid

Never let them know your next move: Take W

Another dido video, we sleep tonight lads!!!!!!

I'm not a big fan of maths. But Dido maths on the other hand 👌

One envelope has x and the other 2x. Or you can say one envelope has x and the other x/2. Which of these two ways of putting it that you choose doesn't matter, they amount to the same thing. One envelope has twice the other, or you can say one envelope has half the other. But as soon as you describe and mark the contents of one envelope as "x" and the other as "x/2 or 2x", then you run into a problem. You conjure up three sums of money into play where before there were only two. Of these three, the biggest is 4 times the smallest, where before the maximum ratio was 2. No wonder confusion takes over. Yet nobody ever seems to question this particular move. One way out is to say you want three sums? OK so let's have a three envelopes game, one for each sum. The x envelope is marked "x", but the other two are both marked identically as "x/2 or 2x", though one only has x/2 while the other has 2x. If you have the x, then a moment's reflection will tell you it indeed makes sense to switch it for either one of the other two envelopes at random, since you stand to win 2 times what you stand to lose with the same probability. We can now see how two different games were being confused with each other.

do you know about the monty hall problem?

I pick B every time because I am too simple to be affected by paradoxes.

switching is a lose and win...

Like monte hall

I hate this paradox I hate this paradox so much it’s so stupid (I am a math major) great asmr though!!! I would love to keep watching but this paradox makes me unreasonably angry and i need to get to sleep

great video but absolutely brutal way to write an x

so it's basically the Monty hall problem?

Everyone who loves dido and has been watching them for a while 👇

the discord link is invalid.

Am I the only one who saw that thumbnail and immediately thought this was a super old Chris Stuckman video?

learned this from the movie 21

infinite money glitch

I actually chose B

You use dollars in Scotland? Wow

Really sucks to have mid rolls in an asmr video

Thank you for actually knowing this unlike everyone else that I tried to explain this to in school You are not part of the dumb group in my standards Not that anyone will care about my opinion because one person against thousands of other commenters my comment will probably not be seen

I’ll stick with my 50 50 chance thank you very much 😂

It is not that complicated, just pick an envelope

Stop wearing makeup dude