
ball carton
Any help offered in solving the below question will be much appreciated.
Thanks
Kingman
A manufacturer makes balls which are packed in cartons of 20 and send to market
for sale. A ball will not meet the minimum requirement for packing for sale if it weighs less than 20 grams. On average 2% of the balls made did not meet the minimum requirement.
Find the probability that out of 4 randomly chosen cartons of balls, there are exactly 2 cartons with at least 1 ball that does not meet the minimum requirement. (0.295)

Well, there are a couple of things that you have to do here.
First, we must find the probability of at least 1 ball in a carton of 20 weighing less than 20g.
We can use a binomial distribution for this. Here's the formula.
$\displaystyle pr(X=x)={k\choose x}p^{x}(1p)^{(kx)}$
X~binomial(k=20,p=0.02)
X= the number of balls that weigh less than 20g.
We're looking for $\displaystyle pr(X\geq 1)$
Using the Axioms of Probability this can be rewritten as $\displaystyle 1  pr(X<1)$ which is the same as $\displaystyle 1pr(X=0)$
Using the binomial distribution formula given above solve
$\displaystyle 1pr(X=0)$
Your answer (I'll call it y) will be used in the second part.
Now we need to find the probability that out of 4 randomly selected cartons, exactly 2 will have at least 1 ball that does not meet the minimum weight requirement.
Again, we can use a binomial distribution.
X~binomial(k=4, p=y)
X= the number of cartons that have at least 1 ball that does not meet the minimum weight requirement.
Now simply use the binomial distribution formula to solve
$\displaystyle pr(X=2)=...\approx 0.295$