Welcome to my third video! This is exploring non-transitive of probability distributions with the aid of Efron’s dice. Thanks to Hayley for joining me in this one! As always, any suggestions on future topics are welcome. I have plenty of subjects to come - the ideas come much quicker than I can actually make the videos. Anyway, any good ideas will be added near the top of the list.
Great video! Really liked some of the visuals and hope you make more videos! Couple things I noticed that could make it even better: 1) I think the animation you had for the 73% example was really great and I visually got the idea of what was happening, but I didn't understand where the calculations were coming from - maybe that part could have been assigned to a companion video / appendix, since it was interesting but not crucial to the rest of the video. 2) A small editing note - the music at the start and the end is great, but it's a little loud as you start speaking. Your audio is a lot better than in your first video, and it feels like you're more comfortable on camera which is great! Regarding Arrow's impossibility theorem, I honestly think the statement of the theorem is more interesting than the proof - I'd prefer videos more about 'paradoxes' / counter-intuitive probability facts, but that's just my opinion :) Once again love the vid!
It is possible to make the 4 intransitive dice example but instead keeping equal the variance of all dices? I found really suspicious the results you found experimentally on both cases
One of the dice has zero variance so, can’t make the same. Generalising to probability distributions (or n-sided dice) you could, for winning probabilities less than 2/3. Not sure what it would prove though. What did you find suspicious? The winning margin in the example games? Yes, that was surprisingly large. I did do other games (rejected because camera/lighting setup was wrong) where it was closer to what you would expect. Just, the final run through and Hayley won almost all the dice rolls!
@@almostsure I already comment on other video about how different variances would change the chances of winning a race of 3 or more dogs with speeds distributed normally with same expected values, differently from the 2 dogs scenario where variance is irrelevant. My intuition says that the game you did you could have triggered the same kind of asymmetry: if you choosed first a dice an the other choose later a "winner" one and don't change again you know you will lose/win on average, but when you started to change dices by randomly picking the first choice a random number of times, probably you skewed in different ways your distributions: Imagine a new game by 4 people: 2 of them plays exactly the game you did but randomly peaking only the two dices with higher variances, and by default the other with the remaining low variance dices. You started the game and the second group the first person only choose the dice with only 3s so its distribution is known... you will expect a very different distrubution than the other randomly peaking group which distribution will be the convolution of two wide variance distributions, so their distribution should be even wider. That is interesting about you first game. I think by randomly picking dices you introduced another asimmetry in your experiment additional to the intransitive dices phenomena. Its an interesting review
Alexandra N. Yakusheva (2022). Nontransitive dice with equal means and variances. English abstract. This study aims to investigate the nontransitivity of the stochastic precedence relation. The dice were taken as an example of discrete random variables with a finite set of values. The means and variances of the dice were assumed those of the classical dice. The whole variety of nontransitive sets containing three or four dice were found in case of one or two tosses and various ways to determine the advantage. The sets that reveal the strongest property of nontransitivity were obtained according to the specific function. The hypothesis has been tested about the emergence of non-transitivity after two tosses of dice in originally transitive sets. The main text is in Russian.
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Welcome to my third video! This is exploring non-transitive of probability distributions with the aid of Efron’s dice.
Thanks to Hayley for joining me in this one!
As always, any suggestions on future topics are welcome. I have plenty of subjects to come - the ideas come much quicker than I can actually make the videos. Anyway, any good ideas will be added near the top of the list.
The second I saw your channel was about statistics and probability theory I subscribed so hard I dropped my phone
Hayley's luck stats through the roof xD
You’re not wrong!
Hi
@@diamonddiamonds7791Hi Hayley!
Great video! Really liked some of the visuals and hope you make more videos! Couple things I noticed that could make it even better:
1) I think the animation you had for the 73% example was really great and I visually got the idea of what was happening, but I didn't understand where the calculations were coming from - maybe that part could have been assigned to a companion video / appendix, since it was interesting but not crucial to the rest of the video.
2) A small editing note - the music at the start and the end is great, but it's a little loud as you start speaking. Your audio is a lot better than in your first video, and it feels like you're more comfortable on camera which is great!
Regarding Arrow's impossibility theorem, I honestly think the statement of the theorem is more interesting than the proof - I'd prefer videos more about 'paradoxes' / counter-intuitive probability facts, but that's just my opinion :)
Once again love the vid!
Thanks for the comments!
Good. Careful with the gambling.
Hi I love your video especially if I am in a vid ❤❤❤❤❤
Thanks for joining me in this one!
It is possible to make the 4 intransitive dice example but instead keeping equal the variance of all dices? I found really suspicious the results you found experimentally on both cases
One of the dice has zero variance so, can’t make the same. Generalising to probability distributions (or n-sided dice) you could, for winning probabilities less than 2/3. Not sure what it would prove though.
What did you find suspicious? The winning margin in the example games?
Yes, that was surprisingly large. I did do other games (rejected because camera/lighting setup was wrong) where it was closer to what you would expect. Just, the final run through and Hayley won almost all the dice rolls!
@@almostsure I already comment on other video about how different variances would change the chances of winning a race of 3 or more dogs with speeds distributed normally with same expected values, differently from the 2 dogs scenario where variance is irrelevant.
My intuition says that the game you did you could have triggered the same kind of asymmetry: if you choosed first a dice an the other choose later a "winner" one and don't change again you know you will lose/win on average, but when you started to change dices by randomly picking the first choice a random number of times, probably you skewed in different ways your distributions:
Imagine a new game by 4 people: 2 of them plays exactly the game you did but randomly peaking only the two dices with higher variances, and by default the other with the remaining low variance dices. You started the game and the second group the first person only choose the dice with only 3s so its distribution is known... you will expect a very different distrubution than the other randomly peaking group which distribution will be the convolution of two wide variance distributions, so their distribution should be even wider.
That is interesting about you first game. I think by randomly picking dices you introduced another asimmetry in your experiment additional to the intransitive dices phenomena. Its an interesting review
Alexandra N. Yakusheva (2022). Nontransitive dice with equal means and variances.
English abstract. This study aims to investigate the nontransitivity of the stochastic precedence relation. The dice were taken as an example of discrete random variables with a finite set of values. The means and variances of the dice were assumed those of the classical dice. The whole variety of nontransitive sets containing three or four dice were found in case of one or two tosses and various ways to determine the advantage. The sets that reveal the strongest property of nontransitivity were obtained according to the specific function. The hypothesis has been tested about the emergence of non-transitivity after two tosses of dice in originally transitive sets.
The main text is in Russian.