Yes. The politician wins. Focus groups and polls are clear on this, they won't vote for random people. Trump would also beat them easily.
(Obviously some probability you get someone effectively not random from a random draw, but it's very low.)
I'm interested in the probability. Sorry for giving the impression I wanted a yes or no answer; I've edited the title accordingly.
ETA: Another way of putting my question is: How many eligible Americans are there who, if they magically ended up running against the current president in a 1v1 race, would win? Is it zero? Is it ten thousand? Is it a million? Ten million?
Random person most likely has an IQ of 100 which is a standard deviation or two below a random president. Random person most likely has less talent and experience at politics, promotion and attack than a politician. Random person is most likely less physically attractive, less charismatic, and less wealthy than a politician. Random person doesn't have a gigantic support apparatus behind them (although even if they do they're probably still screwed because it's not enough to make up for the rest of the deficiencies). As others say it won't even be close. Random person will predictably be slaughtered.
Not a very principled answer, but: 98%
There is already a substantial preference for incumbents (about 65-35 I think), and I think this would be much stronger if the challenger was completely unaffiliated with politics (I want to say something like 90-10 if challenger and sitting president were equally electable otherwise, maybe 75-25 if the challenger is actually substantially better than the president, 99-1 if they're just average).
Say there's 5% chance they're equally or more electable, 2% they're substantially more electable. Then there's 0.5% on them being equally electable and winning, 0.5% on them being substantially more electable and winning and 1% on them being less electable but winning anyway.
So 98% overall.
Edit: didn't consider that the major party might organise in support of the candidate, this changes things a lot. Say there's a 20% chance that the out-of-power party campaigns for the rando (40% the rando is somewhat-aligned, 50% conditional that the party can get their shit together to campaign for them); then the incumbency advantage will be more like the standard 65-35, assuming equal presidentiality (but I still think they're probably a lot less presidential)
All told...this gets me to 96% chance the sitting president wins.
You are the only person who has actually answered the question; thank you.
The thing that interests me about this is that it seems like if you are right, both major parties are doing a terrible job of selecting candidates to run against the sitting president. There seem to be millions of people who would win. Maybe they care about more than just winning the election? Or maybe even though there are millions of people who would win, there's no one who has a legibly higher chance of winning than [whatever candidate the GOP ends up selecting?]
Politicians are good at campaining. Most people are not. A moderately successful politician at the national level would shred through any random people.
I disagree with this. At the president level, politicians are more like a brand, than a source of labor. Most of the work is done by specialists; all they need is good acting skills, and specialists are good enough to figure out all sorts of ways to make the random person play to their strengths on the podium (there would even be game theory between them and the politician during the debate).
To add to the consensus— if a random person actually had a favorable matchup against a career politician of any stripe, that would be a massive low-hanging fruit for existing political actors to capitalize on. The RNC and DNC would be falling all over themselves to present "average joe" candidates if doing so provided a consistent advantage. They're not, so it follows that either those organizations are both highly un-optimized for winning elections (probably not; too much money at stake) or else that the evidence doesn't bear out that they should.
I don't buy this. I think RNC and DNC are not just trying to win votes; the individuals within those orgs are mainly trying to preserve and increase their own power, which entails winning votes, but it also entails a bunch of internal coalition stuff.
Grabbing an excelent non-politician, and grooming him or her for power, and then attempting to put him in a position of power, without "climbing the ranks", isn't obviously in the interest of any of the party officials of either party.
The VAST majority of adult citizens would decline rather than suffering the scrutiny of the public, and the unpleasant work of internalizing and repeating the lies needed to win, especially with the expectation that it would get even worse if they actually won.
General Sherman had this one right.
There's enough magic/counterfactual assumptions in the question that I don't think I can provide a probability - it's a different universe than I live in, so presumably has different voters and different populace.
Current president of Poland went from being relatively unknown parliament member (I'd bet, less than 5% of general public knew who he was? Also there's another public figure with the same last name, so maybe even some people who knew the name were confusing him with the other guy) to being the president within one year. So with longer timescale, big support from one of leading political powers, and some political backlog it is possible. There are surely similar stories around the world, of outsiders rising to leaders. Of course what you're asking is much more extreme, but that provides some staring datapoint.
Even an unknown member of parliament has still been tested against a competitive market, has at least met many or all the key power brokers, etc. They're much closer to the president end of the spectrum than the random citizen end.
These answers don't take into account the possibility that the party not in power, organizes to support the random candidate.
The political science keywords here I would be looking at are 'polarization', 'slate voting', 'split ticket' and 'downballot/downticket'. Essentially, how many voters are voting for X solely and completely because X is now their party's nominee and that's that, they neither know nor care anything about the candidates, because you simply can't let the Y party win? Rapid career rises are thought-provoking but too qualitative (who ever heard of Barry Obama before he gave that speech and began his run for president? Or imagine if John McCain had died a few years earlier - who outside Alaska, a tiny state population-wise, not even 0.8m people, had ever heard of Sarah Palin? I sure hadn't, but unclear what that means). This can be quantified by looking at the increase in straight party voting, and also by natural experiments: obviously, the presidential candidate hasn't died during campaigning recently so no examples there, but there are a lot of elections out there and many candidates will die or otherwise be incapacitated, and their replacement will be effectively an unknown and probably not all that heavily selected in terms of a political career.
Suppose we magically intervene on the USA to make there be a snap election for the office of President. The incumbent (the existing president) goes up against X, where X is a randomly selected eligible person (e.g. adult, citizen, etc.) Voters have one month of campaigning before they decide, so X isn't a total stranger by the end.
What is the probability that the President wins?
Is the probability substantially different for Biden than it was for Trump? What about Obama?
What is the probability that the President loses in a massive landslide (getting e.g. only 40% of the vote or less)?
ETA: I've changed the title to stop giving the impression that I am looking for a yes or no answer. I'm not looking for a yes or no answer, I'm looking for a probability.