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.
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).
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
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.
That's what I love about these videos! You don't have to enjoy maths or eve understand the maths done in the video - you can still enjoy the ASMR aspect and relax or sleep :)
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.
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.
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
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
The first half of the video purposefully uses incorrect statistics to create a paradox. It doesn't make a difference. Both terms are incorrect for the 2x and the 1/2x which generate the 5/4x. The important part for why it's wrong is shown in the probability tree section. The important distinction with swapping is we are trying to see the net change in value multiplied by the probability of it occurring. For example if you get if you had x and swap to 2x is not 2x but rather +x for value cause your value increased by x and this happens in the 1/2 of the time where you selected x and swapped. If you has 2x and swap you lose -x and so it should include a negative sign for change in expected value and this happens the 1/2 of the time where you selected 2x initially. Thus it ends up being ( 1/2 * x ) + (1/2 * -x) = 1/2x - 1/2x = 0 so no expected change in ev to swap. This matches our initial expectation where each envelope should be worth 1.5x and as such a swap from 1.5x to 1.5x shouldn't change our expected value.
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
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
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25/16 is equal to 5/4. It's not greater than 5/4
@@meankat38💀
@@meankat3825/16 is not equal to 5/4. You can’t take the square root of the numerator and denominator to simplify a fraction.
This is all I needed after a long day at Uni. Your voice is calming and clear. I’m going to sleep well tonight
Hope uni is going well! Thanks for watching
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.
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.
need more of these type of problems videos!
i love your videos about these kinda problems!! super entertaining and interesting
Thanks so much! Let me know what other problems you'd like to see
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
Today's lesson is: Don't be greedy 😅
😂😂😂
I said right foot creep, ooh, I'm walking with that heater
@@smo0ve209🤣
This seems similar to the Monty Hall problem, with the doors and goats and cars and stuff.
Yeah! I mention it in the video too. Quite similar aspects of math problems… except the issue in this one less concrete!
Me partying at 12 at night when I have to be awake In 6 hours because I guessed B had more money
Spoiler
the video hasn’t even started…. thanks mate
not my fault your going through comments before the videos even started???
@@lachlanrYour comment was on top, TH-cam showed me it before I even opened the comments.
@@ZTGAMEZ my bad bro didn’t realise this comment got any likes lmao let alone 84
more asmr like this !!!! first video i seen on your channel and i love it!
Your whispering was great in this one!
So awesome. I’m not that great at math but i love these interesting concepts.
Not sure if this is a real thing - like an Actual paradox. But I love ASMR that I feel like I’m learning something 😂
Me not realizing this channel is ASMR maths 🤣🤣🎉
switching is a lose and win...
Relaxing. I first thought it was wrong all the way but you said it in the end so 👍
Doesn’t matter for me. 10 bucks are 10 bucks I was happy that I even get something 😂
😆
i love your asmr math channel. i also like your British TH-cam name and accent. the British tends to call mathematics maths in abbreviated form.
Dido could you please make a video teaching fractions and %?
Great video! I'd think another fascinating probability problem to cover would be the sleeping beauty problem if you can.
Like monte hall
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.
infinite money glitch
Can you do a video explaining the “Birthday paradox”
Is that the one about 2 people having the same birthday if a certain amount of people are selected at random? 🤔
@@ASMRMaths ya exactly. It’s if your in a room with 23 random people you have a 50% chance of sharing the same birthday with someone else in the room
I love how the shape of the mic ( I think) looks like a potted tree
Does it? Haha
This is great. Nice to meet you
I'm not a big fan of maths. But Dido maths on the other hand 👌
That's what I love about these videos! You don't have to enjoy maths or eve understand the maths done in the video - you can still enjoy the ASMR aspect and relax or sleep :)
To be honest I think I am starting to understand maths more because of these videos. Thanks Dido 👍
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.
I pick B every time because I am too simple to be affected by paradoxes.
do you know about the monty hall problem?
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.
Hi there! I recently did a trigonometry video ☺️ check my latest videos
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
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
The first half of the video purposefully uses incorrect statistics to create a paradox. It doesn't make a difference. Both terms are incorrect for the 2x and the 1/2x which generate the 5/4x. The important part for why it's wrong is shown in the probability tree section. The important distinction with swapping is we are trying to see the net change in value multiplied by the probability of it occurring. For example if you get if you had x and swap to 2x is not 2x but rather +x for value cause your value increased by x and this happens in the 1/2 of the time where you selected x and swapped. If you has 2x and swap you lose -x and so it should include a negative sign for change in expected value and this happens the 1/2 of the time where you selected 2x initially. Thus it ends up being ( 1/2 * x ) + (1/2 * -x) = 1/2x - 1/2x = 0 so no expected change in ev to swap. This matches our initial expectation where each envelope should be worth 1.5x and as such a swap from 1.5x to 1.5x shouldn't change our expected value.
great video but absolutely brutal way to write an x
Never let them know your next move: Take W
😂
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
More paradox asmr pls
Everyone who loves dido and has been watching them for a while
👇
the discord link is invalid.
It is the stuff that I really like 😊
so it's basically the Monty hall problem?
Not quite… but it uses a similar concept!
@@ASMRMaths ah ok cool
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
Another dido video, we sleep tonight lads!!!!!!
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
It’s such a fun problem! I hope you liked it. Thanks for your comment 🙂
I actually chose B
You use dollars in Scotland? Wow
We don’t! I just use it as majority of my audience are from the United States
Really sucks to have mid rolls in an asmr video
I’ll stick with my 50 50 chance thank you very much 😂
It is not that complicated, just pick an envelope
You know nothing
Stop wearing makeup dude
I don’t 🙂