Touché. I was thinking a decent salary to be over 70k/yr based on some article I read about satisfaction vs earnings.
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I’m not a mathematician, but that looks like a normal distribution bell curve. For this scenario, the distribution is bimodal ($0 or $50 million). Thus, a mean of $25 million is not an appropriate measure of central tendency and is highly misleading. For a normal bell curve, about 2/3 of people would get roughly $25 million (give or take 1 standard deviation). In this hypothetical, literally no one gets $25 million. It’s either $50 million or nothing.
To draw more parallels…
Let’s say there is a country where half of the population dies at birth and the other half live to be 100. Saying that the mean life expectancy is 50 is misleading.
Or a country where half of people are in poverty and the other half are millionaires. Saying that the country is typically middle-class belies the truth of economic inequality.
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Oops! Just saw this. Apologies in advance for my other comment and any overstep on my part. 
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“Expected value” has a very precise meaning to a mathematician (or more accurately a statistician). What’s perhaps missing here is the definition of the problem in terms of statistics, which leads to the expected value. Statisticians will pick that up without even thinking about it (because it’s such a simple scenario). Read the Wikipedia article on expected value if you want to understand what it means when a statistician says “the expected value of the $50 million button is $25 million”.
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Take the 50/50 and potentially make loads of people happy.
That depends on each one. But the proposal here is to analyze, from purely mathematical arguments, which would be the best option.
To start, let’s go to the probabilities:
Red button or green button are mutually exclusive events, so:
1 + 1/2 (100% chance to win $1 million or 50% chance to win $50 million) or
1 + 1/2 (100% chance to win $1 million or 50% chance to win nothing).
But we want to know what would be the best choice.
Note that $50 million is fifty times that amount. It seems obvious to say this, but also note that this works like a bet: you have a 100% chance of winning $1 million, that means you “have” $1 million. If you give up 49% of those chances to make $1 million, you can make fifty times that amount.
The value gets bigger as you give up 1% (out of 100%) chances of winning $1 million.
That is: for every $1 million more you want to make, you need to “bet” 1% of your odds.
100% - $1 million
99% - $2 million
98% - $3 million
…
91% - $10 million
50% - $49 million
51% - $50 million
Since you “have” $1 million, the green button “will be worth” to you $49 million. But the same button says the prize is $50 million. However, the reasoning is simple: if you need to give up a 1% chance of winning a certain amount in order to have the possibility of winning an extra $1 million, that means the opposite also applies: you can “use” $1 million and “buy” 1% more to earn potential green button money.
So, the green button gives you a 50% chance of winning $50 million, but since you “have” $1 million, you get 1% more chance of making that money.
You need to put the “has” in quotes, because as said, they are mutually exclusive events. You will have $1 million as long as you press the red button (and with that, you will no longer be able to press the green one). We’re talking about potential prizes.
When you hit the green button, you miss out on the right $1 million, so it’s like you trade the $1 million for the possibility of winning $49 million more. Note above that, from the model noted here (bet 1% chance for the possibility of winning $1 million more), you have a 50% chance of winning $49 million.
So your chance of making $50 million is 51% ($1 million or $49 million/0.01 + 0.5). By “betting” 49% of your odds, you risk trying to win $49 million more.
In my opinion, and based on the arguments presented here, it is more prudent to press the red button. The other choice would involve giving up (or “spending”) that $1 million to “buy” just 1% more chances to make $50 million.
This is exactly how to think of it: flip the dialogue on what’s happening. You’re not choosing between $1M and $50M, you’re choosing to rely on basically the flip of a coin to not lose $50M AND the guaranteed $1M you would have had.
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Am I the only one growing suspicious of these button presenters and their “millions”? I say we seize the next one who comes along and name our own terms!
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I’m sure if I had a college education in mathematics that might make sense but the expected value of a button that will give me 50 million 50% of the time is either 50 million or zero.
That button will never ever pay a single individual anything other than 50 million or zero. That’s the expected value for us regular folk.
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Going to agree to disagree on this premise.
If you are proposing that, okay, but that’s not the original proposal. And there’s no requirement for every person encountering the thought exercise to engage with it this way.
No offense intended, and I’m sure you have fun amongst yourselves, but this thread is making me think it would not be very fun to run into a bunch of mathematicians at a party. 
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Ha ha ha ha
I’m sure if I had a college education in mathematics that might make sense
I suspect that’s true, and if you were to do that, you would agree that the expected value (a precise mathematical term) is $25 million.
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but this thread is making me think it would not be very fun to run into a bunch of mathematicians at a party.
Don’t worry. If you plan your party during grant season the expected attendance is only 0.2 mathematicians
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But do they tip well?
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For the record, I’m very close friends with a mathematician who went the practical route (financial industry) so you’d think she’d be utterly obnoxious to hang with, but in fact when someone that bright has excellent social skills as well, you get the best of all possible worlds.
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Expected value is a statistical decision-making tool, not an outcome.
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1/2 x $ 50 million + 1/2 x $ 0 = $ 25 million?
True or false? That button is not going to pay any individual anything other than 50 million or zero for any given press?
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As a mathematician, surely you mean ha 4, no?
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