There is a group of 10 medicine sellers in a neighborhood with 80 houses. Each seller is responsible for 8 different houses. The probability to sell medicine to a specific house is 0.8. The seller is excellent if he manages to sell to at least 7 houses. A group is excellent if at least $80$% of the sellers are excellent. What is the probability that in an organization consists of 20 groups of sellers, exactly 2 groups are excellent?
So, I understood it's a Binomial distribution. I think you need to check the probability to be excellent, $P(X=7) + P(X=8)$, from there I can't understand how to distinguish from that the $80%$ of sellers that are excellent. any direction?
The probability that a given individual is excellent is given by ...
$$p_1 = \binom 87 (0.8)^7 (0.2)^1 +\binom 88 (0.8)^8 (0.2)^0 \sim 0.503 $$
The probability that a group is excellent is given by ... $$p_2 = \binom {10}8p_1^8 q_1^2 + \binom {10}9p_1^9 q_1^1 + \binom {10}{10}p_1^{10} q_1^0 \sim 0.057$$
Finally, the probability that exactly 2 out of 20 groups are excellent is ... $$P=\binom{20}2 (p_2)^2(q_2)^{18} \sim 0.215 $$