This problem involves a binomial distribution because we are looking at a fixed number of trials (12 students) with two possible outcomes: attending or not attending the Tet festivities. The probability of success (attending) for each trial is $p = 0.16$, and the probability of failure (not attending) is $1 - p = 0.84$.

Distribution of $X$

Let $X$ be the number of students who will attend the Tet festivities. Since this is a binomial distribution, we can describe $X$ as follows:

$X \sim \text{Binomial}(n=12, p=0.16)$

Here, $n = 12$ is the number of trials (students), and $p = 0.16$ is the probability of a student attending.

Probability that at most 3 students will attend

We need to find $P(X \leq 3)$, which is the cumulative probability of $X$ being 0, 1, 2, or 3.

$P(X \leq 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)$

The probability mass function for the binomial distribution is given by:

$P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$

We will calculate the probabilities for $X = 0, 1, 2, 3$ using this formula and sum them up.

Probability that more than 2 students will attend

This is $P(X > 2)$, which can be calculated as:

$P(X > 2) = 1 - P(X \leq 2)$

Let me calculate both probabilities.The probability that at most 3 students will attend the Tet festivities is approximately $0.8886$, and the probability that more than 2 students will attend is approximately $0.2990$.

Would you like more details on the calculations, or do you have any other questions?

Here are 5 related questions to consider:

1. How would the probabilities change if the number of students surveyed increased to 20?
2. What is the expected number of students who will attend the festivities?
3. What is the standard deviation of the number of students attending?
4. How does changing the probability of attending to 20% affect the results?
5. What is the probability that exactly 4 students will attend?

Tip: When dealing with binomial distributions, remember that you can easily compute cumulative probabilities using binomial tables or statistical software like Python's scipy library!