Many products can operate if, out of $n$ parts, $k$ of them are working. Say $n = 20$ and $k = 17$, $p = 0.05$ is the probability that a part fails, and assume independence. What is the probability that the product operates?
So, I'm a little confused as to why this is $P(X\geq 17)$ instead of $P(X\leq 17)$. Since there are $17$ parts that work out of $20$ parts; shouldn't the probability that parts operate be translated to "probability of at most $17$ parts working", meaning $X$ is less than or equal to $17$? I mean... this question is looking at sample of $20$ parts and $17$ of them are said to be functioning; so it doesn't make sense to say that at least $17$ of them are working (which implies that $17$, $18$, $19$, or $20$ parts are working) since $17$ parts are the maximum parts that could operate.
"I mean... this question is looking at sample of 20 parts and 17 of them are said to be functioning; "
I think there is some misunderstanding here. The question meant to say that out of 20 parts, if (at least) 17 of them are functioning, then the product is still working.
So if $X$ is the random variable: Number of parts functioning out of 20, then the required probability should be $P(X\geq 17)$.