Question

A higher education research Institute at UCLA collected data from 203,967 incoming first time full-time freshman students from 274 year colleges and universities in the US 71.3% of those students reply that yes they believe that same sex couples should have the right to legal marital status suppose you randomly pick six first time full-time freshman from the survey you are interested in the number that believe the same sex couple should have the right to legal marital status what is the probability that at least two of the freshman reply yes

Answer 1

This is a binomial distribution problem with n = 6, p = 0.713, and q = 1 - p = 1 - 0.713 = 0.287. We want to find the probability that at least two of the six freshman reply "yes."

Using the binomial probability formula, we can calculate the probability of each possible outcome:

P(X = 0) = (6 choose 0) * 0.713^0 * 0.287^6 = 0.0022

P(X = 1) = (6 choose 1) * 0.713^1 * 0.287^5 = 0.0262

P(X = 2) = (6 choose 2) * 0.713^2 * 0.287^4 = 0.1283

P(X = 3) = (6 choose 3) * 0.713^3 * 0.287^3 = 0.2822

P(X = 4) = (6 choose 4) * 0.713^4 * 0.287^2 = 0.3076

P(X = 5) = (6 choose 5) * 0.713^5 * 0.287^1 = 0.1786

P(X = 6) = (6 choose 6) * 0.713^6 * 0.287^0 = 0.0458

To find the probability that at least two of the six freshman reply "yes," we need to add the probabilities of the outcomes where X is greater than or equal to 2:

P(X >= 2) = P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6)

= 0.1283 + 0.2822 + 0.3076 + 0.1786 + 0.0458

= 0.9425

Therefore, the probability that at least two of the six freshman reply "yes" is approximately 0.9425, or 94.25%.