Answer
14
Work Step by Step
Let $X$ be the random variable of number of cameras failing the test. $X$ has the binomial distribution with parameters $n, p=1-0.8=0.2$
The probability mass function of $X$ is given by:
$$
\mathbb{P}(X=x)=\left(\begin{array}{l}
n \\
x
\end{array}\right) 0.2^{x} \times 0.8^{n-x}, x=0,1, \ldots, n
$$
Calculate using this formula:
$$
\begin{array}{l}
\mathbb{P}(X=0)=0.8^{n} \\
\mathbb{P}(X \geq 1)=1-\mathbb{P}(X=0)=1-0.8^{n}
\end{array}
$$
We are looking for the smallest $n$ such that
$$
\mathbb{P}(X \geq 1) \geq 0.95
$$
This leads to:
$$
\begin{array}{c}
0.8^{n} \leq 0.05 \\
n \ln 0.8 \leq \ln 0.05 \\
-0.22 n \leq-2.99 \\
n \geq 13.61
\end{array}
$$
Therefore we conclude that the smallest sample size is $n=14$
