Context
I ran into this problem in the book Fundamentals of Probability with Stochastic Processes (problem 5.1.7): "A manufacturer of nails claims that only 3% of its nails are defective. A random sample of 24 nails is selected, and it is found that two of them are defective. Is it fair to reject the manufacturer’s claim based on this observation?"
The book provides the following answer: "It is not fair to reject the company's claim since there is a 16.2% chance that two or more defective nails are found."
A more intuitive answer, to me, is: "It is not fair to reject the company's claim since there is a 12.71% chance that exactly two defective nails are found."
My Solution
Let $X \sim Bin(n = 2, p = 0.03)$ be the number of defective nails. Then the probability mass function is: $$p(x)=\binom{24}{x}(0.03)^x(1-0.03)^{24-x}$$
We are interested in $P(X = 2) = p(2) = 0.127095 \approx 12.71$%. So, it is about $12.71$% likely that we observe 2 (of 24) defective nails. As is common, we choose that any observation which is less than $5$% likely to occur is considered "very unlikely" and can thus be rejected. The above observation is not less than $5$% likely, so we cannot reject the manufacturers claim.
My Question
It is weird, to me, that the book chooses the event "at least $2$ defective nails are observed" to describe the probability of the event "$2$ defective nails are observed." I see that it is increasingly improbable that there are defective nails past $2$, but why are we incorporating these later probabilities to begin with? We didn't observe at least $2$ defective nails, we observed exactly $2$. In what case does my solution fail where the other one doesn't?
Ask yourself what are the set of possible experimental outcomes that support the claim of a defect rate that is no worse than $3\%$, versus outcomes that contradict such a claim. Bear in mind that the manufacturer's claim implicitly asserts that the defect rate is no worse than $3\%$, since if it were found through extensive sampling that the defect rate was, say, $0.01\%$, that makes the manufacturer look even better than the claimed defect rate.
Put another way, it's only when the proportion of defects clearly exceeds $3\%$ that we should start to suspect that the manufacturer's claim is untrue--specifically, that it is untrue in a way that does not favor the manufacturer.
In this context, we can see that it is natural to consider the set of outcomes for which there are "too many" defects observed in the sample--i.e., assuming that the manufacturer's claim is true, what is the chance of observing as many defects as you obtained, or more? That is why the solution calculates the probability of at least two defects, and not just exactly two.
For another example, suppose I show you a coin and I assert that it is fair. You inspect the coin and see that there is a heads side and tails side. Based on your inspection you have no reason to doubt that it is fair. So you conduct an experiment and you flip it $n = 50$ times. As a result of your experiment, you obtained $X = 30$ heads and $n - X = 20$ tails. This makes you think it might not be fair, but the difference isn't that huge: after all, a perfectly fair outcome would have been $25$ heads and $25$ tails.
Now, if you calculated the probability of this exact outcome of $X = 30$ heads, assuming a fair coin, it would be $$\binom{50}{30} (1/2)^{30}(1 - 1/2)^{20} \approx 0.0418591.$$ That's only $4\%$. So would you say the coin is unfair? Before you do, you might want to observe that the outcome with exactly $25$ heads and $25$ tails has probability $$\binom{50}{25} (1/2)^{25} (1 - 1/2)^{25} \approx 0.112275.$$ If the coin is fair, why is the probability of equal heads and tails only $11.2\%$? You can start to see that the issue here is that if the coin were fair, you are still hardly assured that you will get an equal number of heads and tails. Some deviation exists by random chance. Conversely, if the coin were unfair, it stands to reason that outcomes that were even more extreme than the one you had seen ($30$ heads and $20$ tails) should also be counted when computing the probability associated with the evidence in favor of rejecting the claim that the coin is fair.