Psychology / heuristics of $\Pr(A)\le \Pr(B)$ when $A$ is a subset of $B$ A while ago I read about the flaws people have in assigning probabilities to events based on their beliefs and experiences. For some reason, I started thinking about that today, and realized that I don't recall the main points of the argument, and really want to understand it.
I'll try to state it here as I recall it. Do you see any obvious errors in the questions or in my conclusion?
R = all rock stars
M = all rock stars who are Grand Masters (GM) in chess

What is the probability of a rock star to play chess after concert?
Doesn't seem likely, right? Say,
$\Pr(R) = 0.2$
What is the probability of a GM rock star to play chess after concert?
A lot more likely:
$\Pr(M) = 0.8$
But, $\Pr(R)$ must be $> \Pr(M)$ as $M$ is a subset of $R$, right? So one or both of the probabilities has to be wrong. 


*

*So, does this mean I have to modify $\Pr(R), \Pr(M)$ or both?

*Or in order to have said $0.2$ I had to know the size of $M$?

*Or is there a problem with how the question is stated?


Also, if you can point me to a book or a paper on the subject, I'd really appreciate that.
 A: 
Do you see any obvious errors in the questions or in my conclusion?
R = all rock stars
M = all rock stars who are Grand Masters (GM) in chess

Starting at the most basic, you should define your events as events (sets of outcomes), rather than sets containing all members for whom the event is true.
Like this:
Let $R$ be the event "is a rock star"
Let $M$ be the event "is a rock star who is a Grand Master in chess"

What is the probability of a rock star to play chess after concert? Doesn't seem likely, right? Say,
Pr(R)=0.2

The problem right here is you have changed the meaning of the event R. You're trying to talk about the probability that a rock star plays chess after a concert. That is, you need to define a new event:
Let $C$ be the event "plays chess after a concert"
Then you can properly define your probability, which I presume to be a conditional probability:
$P(C|R)=0.2$
"Given someone is a rock star, the chance they play chess after a concert is 0.2"
This is easier to read left to right if you say it as:
"The chance someone plays chess after a concert given they're a rock star is 0.2"
(it's a little awkward phrased that way in this case, because we introduce 'after a concert' before we know this is someone for whom that's even possible, which makes the English phrasing seem odd. Usually it works just fine this way though.)
Similarly
$P(C|M)=0.8$

But, Pr(R) must be >Pr(M) as M is a subset of R, right?

Right, and now it can be, because we're not confounding it with something else.
You can, for example, talk about $P(M|R)$ being small, or that $P(M\cap C)<P(R\cap C)$
($\cap$ works like it always does with sets, being an intersection; read it out as 'and'; many people write that instead as $P(M C)<P(R C)$)
$ $
You have to be very careful about defining your events and distinguishing conditional probabilities (like $P(C|M)$) from joint probabilities (like $P(C\cap M)$).

As for the psychology of people's assessment of such things, people seem to think in terms of stories, or scenarios.
The story of "a rock star grand master who plays chess after a concert" works as a tale.
The story of "a rock star who plays chess after a concert" doesn't.
This causes people to tend to internally (but not explicitly) confound the concept relating to the conditional event $(C|M)$ with the one relating to the compound event $(CM)$ and hence regard $P(CM)$ as more likely than $P(CR)$ even though the event $CM$ is a proper subset of $CR$.
