I have some probability questions that I'm struggling with and would appreciate some help. I included my attempt at a solution below each question.
When circuit boards used in the manufacture of compact disc players are tested, the long run percentage of defective is $5\text{%}$. Let $X=\text{# defective boards}$ in random sample of size $n = 25$, so $X\sim\text{Bin}(25, 0.05)$.
Find the probability that the number of defective boards is:
a) At least $5$:
$$Pr(X\geq 5)=1-(\Pr(X=0)+\Pr(X=1)+\Pr(X=2)+\Pr(X=3)+\Pr(X=4))$$
$= 1 - 0.9928 = 0.0072$
But the teacher's solution says the answer is $0.993$ and I'm assuming they did what I did, but didn't subtract the numbers from $1$. I don't understand why because I thought I would be counting every value when $X$ is $5$ or greater than that? For questions like $\Pr(X\leq 5)$ I would solve that by adding $\Pr(X=0)+...+\Pr(X=5)$ so I don't understand why I'm supposed to basically do the exact same thing even though the signs are flipped?
b) Between $1$ and $4$, inclusive.
For this question, I figured I would do:$\Pr(1\leq X\leq 4) = \Pr(X=1)+...+\Pr(X=4)$
But that gives me $0.9928$ which is what I was supposed to get for part (a). The teacher's solution, however, says the answer is $0.715$ and I don't understand why.
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