Conditional Probability (The cookie Poroblem) Gentlemen, 
I have small confusion in finding donditional probability in the "Cookies Problem" describe below: 
Suppose there are two full bowls of cookies. Bowl #1 has 10 chocolate chip and 30 plain cookies, while bowl #2 has 20 of each. Our friend Fred picks a bowl at random, and then picks a cookie at random. 
If I ask, what is the probability that Fred picks a plain cookie given Bowl 1?
People say the right answer is straight forward: 
$
P(\text{Plain|Bowl1}) = \frac{30}{40}= \frac{3}{4}
$
But I'm confused, why we did not invoke the probability of choosing Bowl 1? 
I mean, why we did not use the following: 
$
P(\text{Plain|Bowl1}) = \frac{30}{40}\frac{1}{2}= \frac{3}{8}
$
please clarify, 
Thank you so much, 

Intuitively: It is the probability of picking the cookie given that it was bowl one.  You do not include the probability for picking that bowl because it was given that it was picked.
