I'm doing practice problems in my book to help me gain a better understanding.
The amount of cola in a 355 ml bottle from a certain company is a random variable with a mean of 355 ml and a standard deviation of 2 ml. For a sample of size 32, perform the following calculations
a) Find an approximate probability that the sample mean is less than 354.8 ml
P($\bar{X} < 354.8$) = P($Z < \cfrac{354.8 - 355}{2 / \sqrt{32}}) = .2858$ (approximately) which matches the book's answer b) Now suppose the amount of cola is distributed as $N(355,4)$. Find an approximate probability that 10 bottles in the sample contain less then $354.8$ ml of cola.
Ok, I did P($X=10$)= $C(32,10) \cdot 0.2858^{10} \cdot .7142^{22} = .1427$ approximately
But the correct answer is $0.0346$
Why? Can someone help me understand where I went wrong?