Yes, probability measures are unobservable. If you assume that a large number of observations are drawn from the same probability measure you can start to approximate it. But this requires assumptions like time homogeneity and an "ergodic"-type hypothesis, namely that the population statistics are the same as the time statistics of a single instance of the population - not weak hypotheses, but without them you are talking about the probability of single events and then you have a lot of philosophical and practical problems.

Let's look at this probability of a single event in a little more detail... We have a single event, the result of a single football game.
RogueTrooper:

The miserable 3-2 defeat by Croatia at Wembley saw England finish in third place in qualifying Group E and led to the Football Association terminating the contract of head coach Steve McClaren after just 18 games.

This has economic consequences:
ThatBritGuy:

£2bn of domestic spending lost according to some estimates.

even consequences for stock valuation:
Jerome a Paris:

England's spectacular crash out of Euro 2008 on Wednesday night drove Sports Direct to warn that it was unlikely to meet expectations for profits in the current year.

The boot of Mladen Petric, who scored a late strike to lead Croatia to victory against England, also dented the prospects at Umbro.

So, trading Umbro or Sports Direct was in effect a bet on the outcome of the England-Croatia match.

Assume for the sake of argument that the "true" probability (objective-probability-interpretation alert!) of an England win (Achtung! single-event-probability alert!) was 2/3. That is, England was a priori twice as likely as Croatia to win the game. Now, how you would know this is beyond me, but let's assume that God rolled a die with 1-2 giving a Croatia win and 3-6 an England win. But we don't know this.

Now suppose that you're a bookmaker offering fixed odds which you adjust daily so as to make sure you don't lose money, and up to a certain point, 37846 pounds have been bet on an England win and 17364 pounds have been bet on a Croatia win, and you have been continually adjusting the odds as people place bets. How do you set the odds at this point? Note that you're the bookmaker: you're not in the business of betting on the outcome of the match so your opinion of the odds of an Englan win are immaterial to the odds you will offer. Note also that the amount of money bet on either side gives odds of 2.18:1 which overestimates the "true" (unknown by you or anyone) odds of 2:1 (this is presumably the English punters getting moderately carried away by patriotic fervour). So, what odds will you offer?

Using the ratio of money people have already bet with you, and given the fact that 2:1 < 2.18 < 11:5 (continued fractions under the hood, here) you could offer 1:2 odds for Croatia bets and 11:5 odds for England bets. The implied probabilities of 1/3 and 11/16 add up to 49/48 so you're making a profit of 1/48 no matter what. Indeed, if Croatia wins, you have to pay off 17364 * 3 = 52092 and if England wins you have to pay off 37846 * 11/16 = 55049, but people have bet 55230 in total, so if Croatia wins you make 3138 and if England wins you make 181. There is an implicit bet, but you never lose money.

Now, if you happen to have a personal estimate of the odds of the game that falls between 2:1 and 11:5 you won't bet, as the bet is not favourable. But if you believe that the true odds should be 3:1 because you're English or 1:1 because you're Croatian or you have some sort of grudge against the FA or MacLaren, you'll place a bet because it is favourable to you. It is, in fact, highly unlikely anyone will have so precise a bet as to fall betweeen 2:1 and 11:5 so everyone who is inclined to betting is pretty much guaranteed to find the odds advantageous (despite the 1/48 commission that the bookmaker is taking).

In this case, coincidentally, the 2:1 odds offered on one side match the "true" odds that nobody knows about. But note that the calculation of odds and payoffs at no point involves an estimate of the true odds, just an analysis of the market expectation, which is driven in part by irrational considerations and is not very precise (let alone accurate).

Now replace the football match results with the dividends to be given out by a company at the end of the financial year, the bets with buying and selling stocks, the bookmaker with a stock broker (or a "market maker") and the 2:1 and 11:5 odds with the bid and offer for the particular stock. Replace the patriotism or the grudges of the bettors with various cognitive biases of investors big and small, and you'll see the difference between the "market expectation" and the "true valuation" of securities.

The thing is, if you want to get a good estimate of the "true valuation" you need to pay through the nose for computers, analysts,  data, and even then either you have to rely on the judgement of an fundamental analyst or on having good technical analysts, or a good computer "expert system", and half the time you're estimating not fundamentals but the herd instinct of the market participants.

Now, I understand why you could observe the risk-free expectation from the market prices, if the risk-free assumption were correct. It was my impression the Black-Scholes model at least used risk-free probabilities because it made calculations easier, not because it really was a good assumption.

Isn't that true of most applied mathematics?

But your post suggests that more intrinsically you can derive the risk-free expectation (or at least the market's opinion of it?) from current prices but not the actual expectation, nor other people's opinion on this. I definitely do not understand this.

The fact is, if you want to say "the futures market is currently pricing oil as if it will not go below $60/bbl in the foreseeable future" or you want to say "based on the Black-Scholes implied volatility of S&P options we're entering a period of low market volatility", you can say that. But that doesn't mean that oil won't go below $60 or that volatility won't be high. It just tells you what the market expects. And, as we know at least since Keynes (who put them at the center of his macroeconomic theory), expectations drive the economy, but don't tell you anything about the fundamentals.

So, if you think you know about the fundamentals you're likely to find a few investment opportunities because the conventional wisdom is not very wise.

We have met the enemy, and he is us — Pogo

Oh, one more thing. The market price is the expectation with respect to the risk-neutral measure. So studying market prices you can only ever get information about the risk-neutral measure, not the "true" measure.

We have met the enemy, and he is us — Pogo