The Higher Education Research Institute at UCLA collected data from 203,967 incoming first-time, full-time freshmen from 270 four-year colleges and universities in the U.S. 71.8% of those students replied that, yes, they believe that same-sex couples should have the right to legal marital status. Suppose that you randomly pick eight first-time, full-time freshmen from the survey. You are interested in the number that believes that same-sex couples should have the right to legal marital status.

Construct the probability distribution function (PDF). (Round your probabilities to five decimal places.)

To construct the probability distribution function (PDF) for this problem, we can model the situation using a **binomial distribution**. 

### Why a binomial distribution?
A binomial distribution applies here because:
- Each student either believes that same-sex couples should have the right to legal marital status or they do not (two possible outcomes).
- The probability of success (believing in the right to legal marital status) remains the same for each student.
- You are picking 8 students randomly, which is a fixed number of trials.

### Parameters of the binomial distribution:
- \( n = 8 \) (the number of trials or students selected).
- \( p = 0.718 \) (the probability that a student believes in same-sex marriage).
- \( q = 1 - p = 0.282 \) (the probability that a student does **not** believe in same-sex marriage).

The binomial probability mass function (PMF) is given by:

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

Where:
- \( P(X = k) \) is the probability of getting exactly \( k \) successes (students who believe in same-sex marriage).
- \( \binom{n}{k} \) is the binomial coefficient, calculated as:

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

We will compute this for \( k = 0, 1, 2, 3, 4, 5, 6, 7, 8 \) to get the full probability distribution.

### Step-by-step calculation for each \( k \):

#### 1. \( P(X = 0) \)
\[
P(X = 0) = \binom{8}{0} (0.718)^0 (0.282)^8 = 1 \times 1 \times (0.282)^8 = 0.000042
\]

#### 2. \( P(X = 1) \)
\[
P(X = 1) = \binom{8}{1} (0.718)^1 (0.282)^7 = 8 \times 0.718 \times (0.282)^7 = 0.000678
\]

#### 3. \( P(X = 2) \)
\[
P(X = 2) = \binom{8}{2} (0.718)^2 (0.282)^6 = 28 \times (0.718)^2 \times (0.282)^6 = 0.005488
\]

#### 4. \( P(X = 3) \)
\[
P(X = 3) = \binom{8}{3} (0.718)^3 (0.282)^5 = 56 \times (0.718)^3 \times (0.282)^5 = 0.027054
\]

#### 5. \( P(X = 4) \)
\[
P(X = 4) = \binom{8}{4} (0.718)^4 (0.282)^4 = 70 \times (0.718)^4 \times (0.282)^4 = 0.086304
\]

#### 6. \( P(X = 5) \)
\[
P(X = 5) = \binom{8}{5} (0.718)^5 (0.282)^3 = 56 \times (0.718)^5 \times (0.282)^3 = 0.181467
\]

#### 7. \( P(X = 6) \)
\[
P(X = 6) = \binom{8}{6} (0.718)^6 (0.282)^2 = 28 \times (0.718)^6 \times (0.282)^2 = 0.264951
\]

#### 8. \( P(X = 7) \)
\[
P(X = 7) = \binom{8}{7} (0.718)^7 (0.282)^1 = 8 \times (0.718)^7 \times (0.282)^1 = 0.233601
\]

#### 9. \( P(X = 8) \)
\[
P(X = 8) = \binom{8}{8} (0.718)^8 (0.282)^0 = 1 \times (0.718)^8 \times 1 = 0.110414
\]

### Probability Distribution Function (PDF):
Thus, the full PDF is:

\[
\begin{array}{c|c}
\mathbf{k} & \mathbf{P(X = k)} \\
\hline
0 & 0.00004 \\
1 & 0.00068 \\
2 & 0.00549 \\
3 & 0.02705 \\
4 & 0.08630 \\
5 & 0.18147 \\
6 & 0.26495 \\
7 & 0.23360 \\
8 & 0.11041 \\
\end{array}
\]

This table shows the probability of getting 0 to 8 students who believe that same-sex couples should have the right to legal marital status.

Would you like a deeper explanation or have any questions on this?

Here are 5 related questions:
1. How can you calculate the expected value of the number of students who support same-sex marriage?
2. What is the variance and standard deviation of this binomial distribution?
3. How would the PDF change if the sample size increased to 10 students?
4. What is the probability of selecting exactly 5 students who support same-sex marriage?
5. Can we approximate this binomial distribution with a normal distribution?

**Tip**: The sum of the probabilities in any probability distribution function must always equal 1.

The Higher Education Research Institute at UCLA collected data from 203,967 incoming first-time, full-time freshmen from 270 four-year colleges and universities in the U.S. 71.8% of those students replied that, yes, they believe that same-sex couples should have the right to legal marital status. Suppose that you randomly pick eight first-time, full-time freshmen from the survey. You are interested in the number that believes that same-sex couples should have the right to legal marital status. Construct the probability distribution function (PDF). (Round your probabilities to five decimal places.)

Explore how to construct a probability distribution function (PDF) using a binomial distribution for data collected from a UCLA study where 71.8% of first-time freshmen support same-sex marriage. Learn key probability concepts and step-by-step calculations for selecting 8 random students.

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