The 50/50 Problem You Won't Believe

Probability is one of the clearest, more straightforward disciplines within mathematics. Unfortunately, almost nothing in mathematics is murkier and more misleading than… probability. WHAT?!

Belgian mathematician Maurice Kraitchik posed a simple question about wagering wallets, and in doing so he revealed a paradoxical mismatch between a raw, indisputable math conclusion and common sense logic. Why does the math say one thing and reality says another? What do we do when the numbers don’t lie, but we know they aren’t telling the whole truth?

Probability density functions. Bayesian subjectivist analysis. You can put it all together and still not be able to square the math with the simple nagging logic that tells you something different.

Maybe there are some antinomical paradoxes we just have to live with. And maybe that’s okay.

*** SOURCES ***

Merryfield, K., Viet, N., & Watson, S. (1997). The Wallet Paradox. The American Mathematical Monthly, 104(7), 647-649. doi:10.2307/2975058

Carroll, M., Jones, M., & Rykken, E. (2001). The Wallet Paradox Revisited. Mathematics Magazine, 74(5), 378-383. doi:10.2307/2691032

Kraitchik, M. (1942). Mathematical Recreations. W. W. Norton.

*** LINKS ***

Vsauce2:

Hosted and Produced by Kevin Lieber

Research and Writing by Matthew Tabor

Editing by John Swan

Huge Thanks To Paula Lieber

#education #vsauce #maths

[tevaron]

There is not actually a paradox here, there are 2 different things being measured: % Chance of Winning and Risk vs. Reward. Neither player has an advantage when it comes to probability of winning but both have an advantage in terms of what they are risking vs what they could win.

It's Probability vs Expected Value.

(edited for typos)

[Thepsicho]

"how can two things have the same probability of occurring, yet occur less often?"
...they don't?
You are not comparing the odds of getting one specific result against another specific result, you are comparing the odds of getting two results (all yes/all no) against all of the other.

[pie_flavour]

Is the issue not just confusing probability with expected value?

[MMM]

Two things: First, per the video, it looks like Kevin is writing that if you win, you win "X + Y". You're only winning Y, since you already had X before playing. Second, if you win, it's because Y is more than X. If you lose, X is more than Y. If they're both random, X is equal to Y, on average. Your expected winnings equation of "Y minus X" should be based on Y and X being equal (having an expected value of zero), rather than where Y is greater than X.

[cantdrawatall]

I'm kinda confused. Does anyone else feel the "having an advantage" logic he explains for the wallets doesn't make any sense or is it just me?

[BiasProd]

So I'm guessing everyone else also realized how confused this point is, right? It's just flat out incorrect from the start and only gets worse. There's no paradox

[YamadaDesigns]

That Yes/No one isn’t a paradox, because it was talking about a specific permutation, not a class of events

[appleciderhorror]

"Imagine I choose 2 people....I give them the option to play a game." Isn't this a movie franchise?

[yugiohsc]

I feel like you made really wild claims about the nature of probability without thinking it through fully. Also the wallet paradox doesn’t really look like a paradox. What you called logic isn’t really the best form of logic you could use in that situation. Being wrong isn’t a paradox

[BinaryReader]

Yeah, but in the logic case, you didnt really ascribe a probability ratio to win or lose. You said each player stands to make more if they win. That's not the same as the probability of winning, because standing to make more doesn't change the probability of the outcome. Where is the paradox?

[lucidmoses]

Your wallet problem is not logical. One is the odds of winning or not. The other is how much you win or not. That's not the same thing. There is no contradiction.

[theunknown]

A lot of this video's logic is faulty, I'm just gonna focus on the most obvious one:
The chance of the coin flipping the same 5 times in a row is the same as being for example YNYNN
Why?
Because you actually want it to flip exactly Y 5 times in the first case, and you want exactly Y, exactly N, exactly Y, and so on for the second case
It doesn't make sense because we "group" the "in a row" ones together, and "mixed" ones together, that way it looks as if there's a higher chance of them flipping mixed because there are many more options compared to and
We overlook that every single option in the mixed group still has to be called a certain way and if you for example expect a YNYNY and you get YNYN, waiting for the fifth one, the odds are still 50% of it flipping either way, even though it's "mixed".
I hope I cleared it up

[jakubblaha]

I still do not understand where the thought of both players having an advantage came from...

[jaqque]

Conflating combinations with permutations / conflating expected winnings with chance of winning. Not a paradox in the slightest. I also don't carry cash, so I can't lose this game.

[MCRuCr]

The essence of the "paradox" is that if you draw two samples from a distribution, one number is ALWAYS bigger than the other (assuming continuous probability functions) BUT the expected value of both values is THE SAME.

[JustMehChannel]

I am so wildly confused by the wallet paradox. I don't get what's paradoxical about it. The way I understand it is that the only way the advantage comes in is a cost of opportunity and nothing else. The chances of winning or losing are more or less 50/50, but when you win you always win more than you lose.

[hammerth]

The logic of the players is flawed because they think in relative terms compared to their wallet content instead of comparing their wallet content to the distribution of all wallet contents.

If your wallet is flush with cash and you lose, all of it is gone. If your wallet only has a few pennies in it but you lose to someone who only carries cards and coupons, you only lose a couple of pennies. The weight of your wins and losses depends on where within the wallet content distribution you are.

If you are very low within that distribution, a loss carries little weight and a win is likely and likely to be big relative to your wallet content. If you are very high within that distribution, a loss carries a lot of weight and a win is very unlikely and even if you win, it is unlikely that you will win big.

Those factors cancel each other out and the game remains fair.

[johni]

To me it just seems like a conflation of two different ways of being favored.
Neither are favored in the probability of winning but they are both favored in the outcome.

[puzzLEGO]

It’s not a paradox. The ‘advantage’ that a player gets is only determined in this video through the fact that if a player picks at random on average they will win more than they lose. So basically just risk and reward

[veertien]

If I am X, I am "betting" X.
So if I lose, my loss is X.
But if I win, my gain is Y, not X+Y.

Idk if the paradox lays somewhere in it, but that graph with "X+Y" doesn't amuse me...