Finding probability of intersection of events I was reading First course in Probability by Sheldon Ross and am stuck at the understanding this simple problem [hence proved my maths is poor :( ]. 

Problem: Celine is undecided as to whether to take a French course or a chemistry course. She estimates that her probability of receiving A grade would be $1/2$ in a French course and $2/3$ in a chemistry course. If Celine decides to base her decision on the flip of a fair coin, what is the probability that she gets an A in chemistry?
Solution given in the book: Let C be the event that Celine takes chemistry and A denote tha event that she receives an A in whatever course she takes, then the desired probability is $P(C\cap A)$, which is calculated as follows:
$$P(C\cap A) = P(C)P(A|C) = \left(\frac{1}{2}\right)\left(\frac{2}{3}\right) =\frac{1}{3}$$

What I didnt understood above are words in bold italics.


*

*What happened to French course? Its never considered in the given solution.

*How any (whatever the course she takes) is considered in solution.

*Also what happened to fair coin? its never considered in the solution.

 A: 1) Well it wasn't because the question was 
'what is the probability that she gets an A in chemistry?' - her taking the french course doesn't affect this event (doesn't have anything in common with the event mentioned above to phrase it in other way)
2 & 3) 
I'll answer these both question at the same time. 
I hope you understand the concept of conditional probability.
Let's name the events.
$A$ is the event that she gets an A grade in any course
$C$ is the event that she takes the chemistry course
$F$ is the event that she takes the French course.
We want to calculate $P(A \cap C)$ - So the event that she gets an A taking the chemistry course. 
From definition of conditional probability we have that:
$P(A \setminus C)$ $= $$\frac{P(A \cap C)}{P(C)}$ 
so $P(A \cap C)$$=$$P(A \setminus C)P(C)$
In order to do that we have to caltulate both terms on the right hand side:
$P(C)$ is the probability she takes the chemistry course, which is determined by flipping the coin so its - $1/2$
$P(A \cap C)$ is the probability she gets an A while taking the chemistry course (which is given in the excercise) - $2/3$
I Think you will take it from here. The crucial thing to understand is that the whole process of her getting an A has to steps:
1) Sophie (or whatever her name was) flipping the coin so she knows what course she will be taking
2) Her actually completing the course and in effect getting a grade. 
A: 1.The solution for this is for when she gets an A in chemistry, not French. (see the last sentence of the question.)


*Again, the question is for when she gets an A, and not restricted to a particular course.

*The fair coin assumption allows us to use the formula in the solution, which simplifies things.
So, it's all about the wording of the precise question, and the phrase "fair coin" is a hint as to which formula to use in the solution.
I hope this was helpful. The ross book is a good book, but definitely not easy...!
