I have a homework question, and I am a bit confused with the wording of the problem. The question reads as follows:

A boy has a color weakness, so he is not good at distinguishing blue and red. There are 60 blue pens and 40 red pens randomly in a box. Given that he picks up a blue pen, there is a 60% chance that he thinks it is a blue pen and a 40% chance that he thinks it is a red pen. Given that he picks up a red pen, there is a 80% chance that he thinks it is a red pen and a 20% chance that he thinks it is a blue pen.

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So here is what I did:

I said let B be the event that a blue pen is chosen and let R be the event that a red pen is chosen. Then we have P(B) = 0.6 and P(R) = 0.4.

Then I said let X be the event that the boy thinks a blue pen is chosen and let Y be the event that the boy thinks a red pen is chosen. Then we have that:

P(X|B) = 0.6

P(Y|B) = 0.4

P(X|R) = 0.2

P(Y|R) = 0.8

And with this information I found:

P(X n B) = (0.6)(0.6) = .36

P(Y n B) = (0.4)(0.6) = .24

P(X n R) = (0.2)(0.4) = .08

P(Y n R) = (0.8)(0.4) = .32

Now the problem has 3 parts.

A) What is the probability that he picks up a blue pen and recognizes it as a blue pen?

B) What is the probability that he chooses a pen and thinks it is blue?

C) Given that he thinks he chose a blue pen, what is the probability that he actually chose a blue pen?

I do not understand Part A, because I am confused between the boy "recognizing" the blue pen and "thinking" that he chose a blue pen.

Also for part C, if I understand correctly I have to find P(B|X) which I know is equal to P(X n B) / P(X), but I don't understand how to find P(X).