Conditional Probability question

[Juanriq]

Homework Statement

Students who actively participate in class are 4 times more likely to get a good grade than those who don't. 15% of students actively participate in class; 20% of students get a good grade. A person did not participate in class, what is the probability that he got a good grade?

Homework Equations

The Attempt at a Solution

Well, I define P(G): student got a good grade; P(P): student participates. Obviously, P(G) = .2 and P(P) = .15. I also know that P(G|P) = 4P(G| not P). I am looking for the conditional probability P(G| not P). I would imagine that I need to find P(G|P) / 4, but this would mean that I need P(G \cap P) P(P) and alas, I do not believe that I have P(G \cap P). ANy ideas? Anything I am missing? Thanks!

 

[jbunniii]

Note that

P(G) = P(G \cap P) + P(G \cap P^c) = P(G|P)P(P) + P(G|P^c) P(P^c)

You know everything except P(G|P) and P(G|P^c), and you know a relation between those two which should allow you to solve for P(G|P).

 

[Juanriq]

I must be doing something wrong... I have

.2 = (.15)P(G|P) + (.85)P(G|P^c) = (.15)P(G|P^c)/4 + (.85)P(G|P^c)

and upon simplification I get P(G|P^c) = .225, but the answer is supposed to be .16.

 

[jbunniii]

Here is what I get. Write x = P(G | P^c). Then

0.2 = (0.15) (4x) + (0.85) x = 1.45x

so

x = 0.2 / 1.45 = 0.1379...

which doesn't match either of your answers.

P.S. You know .225 has to be wrong, since P(good grade) = 0.2 and surely the probability shouldn't go up for students who don't participate!

 

[Juanriq]

Thanks for all the help! It wouldn't be the first typo in the notes...