After many generations, it is known that 75% of students pass certain subject. if a random sample of 40 students is such that it could be consider they have the same characteristics as the ones observed. ¿Which is the probability that at least 50% of them pass the subject?
I would say that $X\sim Bernoulli(.75)$, also I would like to use Central Limit Theorem or any other distribution to approximate the probability of the 40 student sample. Can you help me?
You are right. You could model each student's success as a Bernoulli random variable with $p = 0.75$ probability of success.
Also, embedding the CLT here sounds like a good approximation because it seems exhaustive to compute the following probability $$Pr( \sum_{i=1}^{40} X_i \geq 20)$$ where $X_i \sim B(0.75)$. Equivalently, we can say (dividing $40$ on both sides) $$Pr( S_{40} \geq 0.5)$$ where $$S_{40} = \frac{\sum_{i=1}^{40} X_i}{40}$$ The CLT tells you that if $X_1 \ldots X_{40}$ are i.i.d (which is your case), then $$\sqrt{40}(S_{40} - \mu) \rightarrow \mathcal{N}(0,\sigma^2)$$ where \begin{align} \mu &= E(X) = p = 0.75 \\ \sigma^2 &= \operatorname{var}(X) = p(1-p) = 0.75(0.25) = \frac{3}{8} \\ \end{align} Hence, we get $$\sqrt{40}(S_{40} - 0.75) \rightarrow \mathcal{N}(0,\frac{3}{8})$$ You could choose to standardize as such $$Z = \frac{S_{40} - 0.75}{\frac{\frac{3}{8}}{\sqrt{40}}} \sim \mathcal{N}(0,1)$$ Going back to your probability $$Pr( S_{40} \geq 0.5)$$ All you have to do is standardize now, i.e. $$ Pr(\frac{S_{40} - 0.75}{\frac{\frac{3}{8}}{\sqrt{40}}}\geq \frac{20 - 0.75}{\frac{\frac{3}{8}}{\sqrt{40}}}) = Pr (Z \geq \frac{20 - 0.75}{\frac{\frac{3}{8}}{\sqrt{40}}})$$ and this is the Normal Approximation of the Binomial Distribution.