The binomial distribution is based on yes/no outcomes and does look like a normal curve.
This is based off of many button pushes each of which are called a Bernoulli trial.
The bernoulli distribution (a single trial) might be more what you are envisioning (a peak at 0 and a peak at 1)
However, the expected value for the Bernoulli is still 0.5!
Expected values are a complex bit of stats that isn’t necessarily intuitive. In fact probabilities are rarely intuitive
Consider that the average family has 1.93 children. This isn’t saying anyone has a kid with 7% missing, but rather describing something about the larger population as a whole. If I look at 100 families I expect to see 193 children. And here if I look at 100 50/50 button pushers I expect 2.5 billion in total funds across all of them. So 25 million each.
This group, where the average person has 25 million, is also an excellent example of why we need to understand what a statistic is telling us. For example, someone (guess which party) could intentionally mislead you to beleive that no one is poor because the average person has 25 million.
Expected value applies based on possible values. It works well in dice throwing or roulette. In this case, it does not apply, as $25 million is not a possible value. There are only two values: 0 or $50 million (green button).
On second thought: the green button has two possibilities. 50% of $50 million and 50% of $0
The expected value:
1/2 x $50 million + 1/2 x $0 = $25 million
This means that if four people hit the green button ($25 million x 4 = $100 million), two of them will win $50 million. Another two will lose. If six hits the button, three will win, another three will lose. And so on.
Excellent point. Also consider there are those who had an average salary, who combined the top ramen @lecti mentioned with decades of reducing housing and other expenses to enable investment. Paying yourself first is not easy, but deciding against it early on leaves even fewer choices later. This is why I would choose the million dollars:
I can retire early now, but would use that money to continue/increase helping family members who are just starting out in their careers or elderly and struggling with increasing costs of living despite being debt free. Even without an unexpected windfall, we’ve already handled all manner of unforeseen health, natural disaster, and other crises that folks tend to bring up. We are each others’ safety net if we cannot manage on our own. No one in my family is an island, which is how we’ve manged - despite generations of being overworked and underpaid - to not only survive, but to thrive.
There is a 1/2 probability to win $50 million. So $50 million Pr{win} = 50 million x 1/2 = $25 million.
However, you are not placing a bet in this case (you are not putting in any money to win $50 million). So I don’t see how expected value can be useful.
Imagine that you pay $4 million to be able to press the green button.
The expected value would be:
0.5 x $50 million + 0.5 x (-$4 million) = $23 million.
This would mean that in the long run (if you bet more than once) your bet would win. Because you would be “expected” to “win” $23 million every time you hit the button.
You would only have negative expected value if you bet more than $50 million, which would be silly.
In case you hit the green button or not, you are not betting anything. Also, you could only press once. Expected value is good when you are going to place more than one bet.
For better or worse, mathematics hijacks regular terms for their own ends. Sometimes this works alright and no one gets confused. e.g. rings in algebra vs. rings on your fingers. Other times it ends up confusing matters. Suppose that we had a bunch of possible outcomes {a, b, c, d, …} where each has probabilities {p(a), p(b), p(c), p(d), …}. Then the expected value is ap(a)+bp(b)+cp(c)+dp(d)+… Why choose that? Well, you can think of it as a kind of weighted average. (e.g. when you repeat the process an infinite number of times, averaging the results as you go) You can also think of it as a kind of center of mass in an analogy to physics. What you can’t do is actually expect the expected value. You are never going to get 0mil$*0.5+50mil$*0.5 = 25 mil$. This happens with other multi-modal probability distributions as well. Calling it the expected value makes people think that they can expect it, but you can’t. Oh well.
Personally, as (an increasingly applied) mathematician, I’d choose the $1mil. I could pay off my mortgage with that. The next real quality of life improvement would be to self-fund my research but $50mil is not enough; I’d still have to write grants.
I mean, if it’s two presses, then it feels like a really simple answer: Press green button first to get a chance at higher payout. If it pays out, then press it again. If it doesn’t, take the comfy $1 mil.
The problem with the green button on a single press is just the 50% loss of a guaranteed one mil. If I’ve got a free toss without consequence, sure.
Apparently those are two slightly different things mathematically but I’m not able to get my head round what the difference is.
As best I can tell the expected value is essentially the average value if you project the expected value far enough. But if it’s only a short series, the expected value and the average might be different.
I’m sure one of our mathematicians will be happy to explain exactly why that is completely wrong…
Not a mathematician, but AIUI, the average value pertains to some real, finite set of cases, whereas the expected value pertains to the abstract set of cases.
As in, you roll 2d6 ten times, and get 6, 9, 2, 4, 10, 7, 2, 12, 7 and 9. The average here is 6.8, when the expected value is 7. If you roll perfectly fair dice infinitely many times, it will converge to 7.
Oh well.
