1. ## 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)

2. 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.

$pr(X=x)={k\choose x}p^{x}(1-p)^{(k-x)}$

X~binomial(k=20,p=0.02)

X= the number of balls that weigh less than 20g.

We're looking for $pr(X\geq 1)$

Using the Axioms of Probability this can be rewritten as $1 - pr(X<1)$ which is the same as $1-pr(X=0)$

Using the binomial distribution formula given above solve

$1-pr(X=0)$

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

$pr(X=2)=...\approx 0.295$