Question

Problem

Yoshi is a basketball player who likes to practice by attempting the same three-point shot until he makes the shot. His past performance indicates that he has a


30


%


30%30, percent chance of making one of these shots. Let


X


XX represent the number of attempts it takes Yoshi to make the shot, and assume the results of each attempt are independent.


Is


X


XX a binomial variable? Why or why not?


Choose 1 answer:


Choose 1 answer:



(Choice A)


A


Each trial isn't being classified as a success or failure, so


X


XX is not a binomial variable.



(Choice B)


B


There is no fixed number of trials, so


X


XX is not a binomial variable.



(Choice C)


C


The trials are not independent, so


X


XX is not a binomial variable.



(Choice D)


D


This situation satisfies each of the conditions for a binomial variable, so


X


XX has a binomial distribution

Answer 1

The random variable X is not a binomial variable because it does not have a fixed number of trials; it continues until Yoshi makes a shot, fitting a geometric distribution instead.

Step-by-step explanation:

The random variable X, representing the number of attempts it takes Yoshi to make a three-point shot, is actually not a binomial variable. This is because, for a random variable to be considered binomial, it must satisfy several conditions, including a fixed number of trials, only two possible outcomes (success or failure) per trial, and that each trial is independent and has the same probability of success.

In Yoshi's case, while there are indeed only two outcomes per trial (making the shot or not) and each trial is independent with a constant probability of success (30%), the number of trials is not fixed. Yoshi continues to shoot until he makes the shot. Therefore, X does not have a fixed number of trials; it can vary from one until however many attempts it takes for success, potentially infinite. This scenario actually describes a geometric distribution, where the variable counts the number of Bernoulli trials up to and including the first success. Hence, the correct choice in this case is (Choice B), which states that there is no fixed number of trials, so X is not a binomial variable.