I am attempting the following problem:
Question: 15% of shirts produced at a manufacturing factory are defective, what is the probability that a random sample of 6 shirts from this factory contain:
1) No defective shirts
2) At least 4 defective shirts
Attempt:
- Define X to be the number of defective products.
- P(defective shirt) = .15
Part 1:
n = 6
p = .15
X~B(6,.15)
\begin{align} P(X=0) & = {6 \choose 0}(.15)^0(1-.15)^{6-0}\\ \nonumber &= (1-.15)^{6}\\ \end{align}
Part 2:
n = 6
p = .15
X~B(6,.15)
\begin{align} P(X \geq 4) & = 1-P(X<4)\\ \nonumber &= 1-P(X=3)-P(X=2)-P(X=1)-P(X=0)\\ \nonumber &=1-{6 \choose 3}(.15)^3(1-.15)^{6-3}-{6 \choose 2}(.15)^2(1-.15)^{6-2}\\ \nonumber &-{6 \choose 1}(.15)^1(1-.15)^{6-1}-{6 \choose 0}(.15)^0(1-.15)^{6-0}\\ \end{align}