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. 71.7% 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.
Answer
š Binomial Probability ā Support for Legal Marital Status
š Understanding the Scenario
This is a classic example of a binomial experiment. We’re selecting a random sample of 8 students, and each student either:
- Supports same-sex marriage (success), or
- Does not support (failure).
š Given Data:
- Sample size (n) = 8
- Probability of success (p) = 0.717
- Probability of failure (q) = 1 ā 0.717 = 0.283
- Random variable (X): number of students in the sample who support same-sex marriage
š§ Binomial Distribution Formula
Where:
C(n, k) = number of combinations = n! / (k!(nāk)!)
š What Can Be Calculated:
You can use this binomial setup to calculate:
- P(X = 6): Probability exactly 6 out of 8 support same-sex marriage
- P(X ā„ 5): Probability 5 or more support it
- Expected value: E(X) = n Ć p = 8 Ć 0.717 = 5.736
ā Interpretation:
The expected number of students (on average) who believe in legal marital status for same-sex couples in a group of 8 is approximately:
This means that if you repeatedly picked random groups of 8 freshmen, about 5 or 6 of them would typically support same-sex marriage based on the survey statistics.
š” Tip: Binomial probability is perfect for modeling yes/no responses in surveys. Use a calculator or software for exact binomial values when needed!