hypergeometic distribution

A company wants to buy 100 bags of salt. They take a sample of 10 bags and agrree to buy the consignment if at most 1 bag in the sample fails to meet their quality requirement. The company that sells the salt knows that 10% of the total consignment will not meet the requirements. What is the probability that they will buy the consignment?

if 10% of the consignment will not meet the requirements, that means 10% of 10 will fail. 10% of 10 is 1. So for every 10 bag sample only 1 bag fails. Since they will accept 1 faulty bag would this not mean that the probability of buying the consignment would be 1?

Look up the binomial theorem.
Let p=probability that bag a_i meets the requirement.
Let q=(1-p)=the probability that bag a_i does not meet the requirement.
Let k=the total number of bags that meet the requirement.

Then \(\displaystyle \L {{10} \choose {k}} p^kq^{10-k}\) represents the probability that k bags will meet the requirement.

The sum of all possible values of k is 1:

\(\displaystyle \L \sum_{k=0}^{10} {{10} \choose {k}} p^kq^{10-k}=p^{10}+10p^9q+45p^8q^2+...+10pq^9+q^{10}=1\)

You are looking for the probability that \(\displaystyle k \in \{0,1\}\)

P.S., Does anyone know how to Tex binomials, or arrays, or combinations in this forum?

Type \(\displaystyle { {N} \choose {k}}\) to see \(\displaystyle { {N} \choose {k}\).