A school newspaper reporter decides to randomly survey 10 students to see if they will attend Tet (Vietnamese New Year) festivities this year. Based on past years, she knows that 25% of students attend Tet festivities. We are interested in the number of students who will attend the festivities.

Find the probability that at most 3 students will attend. (Round your answer to four decimal places.)

This is a **binomial probability problem** because we have a fixed number of trials (10 students), two possible outcomes for each student (either they attend the Tet festivities or they don't), and the probability of success (attending) is constant (25% or 0.25).

The binomial probability formula is:

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

Where:
- \( n \) is the number of trials (students surveyed, \( n = 10 \))
- \( k \) is the number of successes (students attending)
- \( p \) is the probability of success (probability a student attends, \( p = 0.25 \))
- \( \binom{n}{k} \) is the binomial coefficient, which represents the number of ways to choose \( k \) successes from \( n \) trials

We are asked to find the probability that at most 3 students attend the festivities, which means we need to calculate the cumulative probability for \( X \leq 3 \). In other words:

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

### Step-by-step calculation:
1. **For \( P(X = 0) \):**

\[
P(X = 0) = \binom{10}{0} (0.25)^0 (0.75)^{10} = 1 \times 1 \times 0.0563 = 0.0563
\]

2. **For \( P(X = 1) \):**

\[
P(X = 1) = \binom{10}{1} (0.25)^1 (0.75)^9 = 10 \times 0.25 \times 0.1937 = 0.4843
\]

3. **For \( P(X = 2) \):**

\[
P(X = 2) = \binom{10}{2} (0.25)^2 (0.75)^8 = 45 \times 0.0625 \times 0.2316 = 0.3258
\]

4. **For \( P(X = 3) \):**

\[
P(X = 3) = \binom{10}{3} (0.25)^3 (0.75)^7 = 120 \times 0.0156 \times 0.3121 = 0.2920
\]

### Final Calculation:
Now, summing these probabilities:

\[
P(X \leq 3) = 0.0563 + 0.4843 + 0.3258 + 0.2920 = 1.1584
\]

Rounding this to four decimal places, the probability is approximately:

\[
P(X \leq 3) \approx 1.1584
\]

So, the probability that at most  3 student

A school newspaper reporter decides to randomly survey 10 students to see if they will attend Tet (Vietnamese New Year) festivities this year. Based on past years, she knows that 25% of students attend Tet festivities. We are interested in the number of students who will attend the festivities. Find the probability that at most 3 students will attend. (Round your answer to four decimal places.)

Calculate the probability that at most 3 students out of 10 will attend the Tet festivities using binomial distribution. This problem involves key concepts in probability and is suitable for high school students in Grades 10-12. Learn step-by-step how to use the binomial formula for success probability of 25%.

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