I am really confused by the following problem and I am not sure if my solution is correct.
Problem: In a shop there are 300 products which have come from 3 factories (A, B and C). The products from factory A are 70%, the products from factory B are 10%, from factory C are 20%. From the products produced in factory A 80% are first class. From the products produced in factory B only 40% are first class and from the products produced in factory C 75% are first class.
a) What is the probability that an arbitrary chosen product is first class?
b) What is the mean value and the standard deviation of the number of first class objects in the shop?
c) Find the probability that the number of first class products is between 210 and 225.
a) The total number of first class products is 0.8*0.7*300+0.4*0.1*300+0.2*0.75*300=225. So the probability that an arbitrary product is first class is 225/300=0.75.
b) Let $X_i$ be the number of first class products from the i-th factory, i=1,2, 3. Let $X =X_1+X_2+X_3$ be the number of first class products in the shop.
$E(X)=E(X_1+X_2+X_3)=E(X_1)+E(X_2)+E(X_3)= 210*0.8+30*0.4+60*0.75=225 $ (because we have independent Bernoulli trials and the mean is np).
$V(X)= V(X_1)+V(X_2)+V(X_3) = 210*0.8*0.2 +30*0.4*0.6+60*0.75*0.25= 52.05$
and hence the standard deviation is $\sqrt(52.05)$.
Are my solutions to a) and b) correct?
How can I solve c)?