 A school newspaper reporter decides to randomly survey 12 students to see if they will attend Tet (Vietnamese New Year) festivities this year. Based on past years, she knows that 16% of students attend Tet festivities. We are interested in the number of students who will attend the festivities. Give the distribution of X. Find the probability that at most 3 students will attend. (Round your answer to four decimal places.) Find the probability that more than 2 students will attend. (Round your answer to four decimal places.)

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!

A school newspaper reporter decides to randomly survey 12 students to see if they will attend Tet (Vietnamese New Year) festivities this year. Based on past years, she knows that 16% of students attend Tet festivities. We are interested in the number of students who will attend the festivities. Give the distribution of X. Find the probability that at most 3 students will attend. (Round your answer to four decimal places.) Find the probability that more than 2 students will attend. (Round your answer to four decimal places.)

Explore how to calculate the probability that at most 3 and more than 2 students will attend Tet festivities using binomial distribution. The problem involves a survey of 12 students with a 16% chance of attendance, providing key insights into binomial probability. Suitable for students in Grades 10-12.

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