Question

Sixty-five percent of people pass the state driver’s exam on the first try. A group of $50$individuals who have taken the driver’s exam is randomly selected. Give two reasons why this is a binomial problem.

Short answer

The two reasons are:

1. In binomial problem, there must be an infinite number of $n$of independent trials and after each trial the outcome may be success or failure.
2. The probability of pass in each state driver's exam in the first try is constant $p=0.65$,then the probability of fail in each state driver's exam in the first try $q=0.35$
   

Step-by-step solution

Content Introduction

We are given

$65\%$of people pass the state driver’s exam on the first try.

A group of $50$individuals who have taken the driver’s exam is randomly selected.

We need to state the two reason that the given information is about binomial problem.

Content Explanation

The two reasons why the given problem is a binomial problem are:

1. In binomial problem, there must be an infinite number of $n$of independent trials and after each trial the outcome may be success or failure. There are $n=50$individuals who took drivers exam where, individuals were randomly selected and the outcome was either pass or fail. Also, individual exam is independent to each other.
2. Another, is that the probability of success $p$is constant at each trial. Therefore, the probability of pass in each state driver's exam in the first try is constant $p=65\%or0.65$. then the probability of fail in each state driver's exam in the first try is $100\%-65\%=35\%or0.35$. Also, if interested in the outcome success that is pass, then we define the random variable as $X=$number of individuals pass the state driver’s exam out of $50$individuals.