 college claims that 70% of students receive financial aid. Suppose that 4 students at the college are randomly
selected. We are interested in the number of students in the sample who receive financial aid.
a. What constitutes a success on a single trial?
𝑥 = ___________________________________________________________________________
b. What is the probability of success on a single trial?
c. Make a table that shows the probability distribution
d. Find the probability that AT MOST 2 of the students in the sample receive financial aid.
e. Find the probability that AT LEAST 3 of the students in the sample receive financial aid.
f. Find the mean and the standard deviation using the shortcut formulas for the binomial distribution:
𝑀𝑒𝑎𝑛 𝑜𝑓 𝑏𝑖𝑛𝑜𝑚𝑖𝑎𝑙 𝑑𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛: E(x)=
Standard deviation of the distribution = 𝜎 =


We can approach this problem as a **binomial distribution** problem because we are given a probability of success (receiving financial aid) and we are interested in the number of successes (students who receive financial aid) in a fixed number of trials (4 students).

### Part a: What constitutes a success on a single trial?

**Success on a single trial:**
- A success is defined as a student receiving financial aid.

So we can define:

**\(x =\)** a student receiving financial aid.

---

### Part b: What is the probability of success on a single trial?

The probability of success (receiving financial aid) on a single trial is given as 70%. Therefore:

**\( p = 0.70 \)**.

---

### Part c: Make a table that shows the probability distribution

This is a **binomial distribution** where:
- \( n = 4 \) (number of trials),
- \( p = 0.70 \) (probability of success),
- \( q = 1 - p = 0.30 \) (probability of failure).

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

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

Where \(k\) is the number of students receiving aid, \(n = 4\), and \(p = 0.70\).

Let's compute the probabilities for each value of \(k\) (0, 1, 2, 3, 4):

\[
\binom{n}{k} = \frac{n!}{k!(n-k)!}
\]

Let’s calculate each probability and create a table.Here is the probability distribution for the number of students receiving financial aid (\(k = 0, 1, 2, 3, 4\)):

\[
\begin{array}{|c|c|}
\hline
k & P(X = k) \\
\hline
0 & 0.0081 \\
1 & 0.0756 \\
2 & 0.2646 \\
3 & 0.4116 \\
4 & 0.2401 \\
\hline
\end{array}
\]

---

### Part d: Find the probability that **AT MOST 2** of the students in the sample receive financial aid.

This means we need to calculate \(P(X \leq 2)\), which is the sum of the probabilities for \(X = 0, 1, 2\).

\[
P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2)
\]The probability that **at most 2** of the students receive financial aid is:

\[
P(X \leq 2) = 0.3483
\]

---

### Part e: Find the probability that **AT LEAST 3** of the students in the sample receive financial aid.

This means we need to calculate \(P(X \geq 3)\), which is the sum of the probabilities for \(X = 3\) and \(X = 4\).

\[
P(X \geq 3) = P(X = 3) + P(X = 4)
\]The probability that **at least 3** of the students receive financial aid is:

\[
P(X \geq 3) = 0.6517
\]

---

### Part f: Find the mean and the standard deviation using the shortcut formulas for the binomial distribution.

- **Mean of the binomial distribution**: \(E(X) = n \cdot p\)
- **Standard deviation of the distribution**: \(\sigma = \sqrt{n \cdot p \cdot (1 - p)}\)

Let’s calculate both.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, you can easily calculate the mean and standard deviation using these formulas:

- **Mean**: \( E(X) = n \cdot p = 4 \cdot 0.70 = 2.8 \)
- **Standard Deviation**: \( \sigma = \sqrt{n \cdot p \cdot (1 - p)} = \sqrt{4 \cdot 0.70 \cdot 0.30} = \sqrt{0.84} \approx 0.9165 \)

Would you like more details on any part? Here are some related questions for further understanding:

1. How do binomial distributions differ from normal distributions?
2. Can you show how to calculate the binomial coefficient?
3. What are real-world examples of binomial distributions?
4. How would the probabilities change if more students were selected?
5. How do we interpret variance in the binomial distribution?

**Tip**: In binomial distributions, increasing the number of trials tends to make the distribution look more like a normal distribution!

College claims that 70% of students receive financial aid. Suppose that 4 students at the college are randomly selected. We are interested in the number of students in the sample who receive financial aid.

a. What constitutes a success on a single trial?

b. What is the probability of success on a single trial?

c. Make a table that shows the probability distribution

d. Find the probability that AT MOST 2 of the students in the sample receive financial aid.

e. Find the probability that AT LEAST 3 of the students in the sample receive financial aid.

f. Find the mean and the standard deviation using the shortcut formulas for the binomial distribution.

Learn how to apply the binomial distribution to solve a problem involving financial aid recipients. This example explores the probability distribution, expected value, standard deviation, and probabilities for at most or at least a given number of successes. Suitable for students in Grades 10-12.

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