Question

Use the following information to answer the next eight exercises: 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 theU.S. $71.3\%$ 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.

What is the probability that at least two of the freshmen reply“yes”?

Short answer

The probability that at least two of the freshmen reply"yes" is$0.999$.

Step-by-step solution

Given information

$$\begin{gathered} n=8, \\ p=0.713, \\ q=1-0.713 \\ =0.287 \end{gathered}$$

Explanation

The probability of at least two successes out of eight trials is :

$$\begin{gathered} P\left(X=x\right)=\left(\begin{array}{c}n\\ x\end{array}\right)p^x{q}^{n-x};0\le p\le 1,x=0,1,2,\dots ,n,q=1-p \\ P\left(X\ge 2\right)=1-P\left(X<2\right) \\ =1-\left[P\left(0\right)+P\left(1\right)\right] \\ =1-0.00096 \\ =0.999 \end{gathered}$$

Thus, the probability that at least two of the freshmen reply"yes" is $0.999$, which is approximately $1$. So, the event that at least two of the freshmen reply"yes" is almost sure.