Chapter 14, Problem 14.13CYS

To determine

To identify: The distribution.

Answer to Problem 14.13CYS

The correct option is (b) $\text{binomial with }n=3\text{ and }p=\frac{1}{4}$.

Explanation of Solution

Reason for the correct answer:

The random variable x represents the number of contaminated eggs. Here, the random sample 3 (n) eggs have been randomly selected. Moreover, given that one out of four eggs $\left(p\right)$ contains the salmonella bacteria.

Also, there are two possible outcomes (choosing the contaminated eggs and non-contaminated eggs) with the probability of success choosing the eggs having the salmonella bacteria is $\frac{1}{4}$ and not consisting them is $\frac{3}{4}$ $\left(=1-\frac{1}{4}\right)$. Thus, x follows the binomial distribution with $n=3$ and $p=\frac{1}{4}$.

Reason for the incorrect answers:

Option (a) binomial with $n=4$ and $p=\frac{1}{4}$.

Here, Larry does not use more than 3 eggs for cooking and he chose only one out of four eggs containing the salmonella bacteria.

Here, the number of eggs chosen randomly is not 4.

Option (c) binomial with $n=3$ and $p=\frac{1}{3}$.

Here, Larry does not use more than 3 eggs for cooking and he chose only one out of four eggs containing the salmonella bacteria.

Here, one out of three eggs is chosen at random. Thus, $p=\frac{1}{4}$ and $p=\frac{1}{3}$.

Thus, the Options (a) and (c) are incorrect.

Conclusion:

Therefore, Larry uses three eggs chosen at random and that has the distribution binomial with $n=3$ and $p=\frac{1}{4}$. Thus, the correct option is (b) $\text{binomial with }n=3\text{ and }p=\frac{1}{4}$.

Statistics Concept Introduction

Introduction:

Binomial distribution:

The random variable x is defined as the number of success observations that follows the binomial distribution if the distribution has fixed number of observations (n) in which all the observations are independent and have two possible outcomes (success and failure). Also, for each observation, p represents the probability of successes.

The probability using binomial distribution is given by,

$P\left(X=x\right)=\left(\begin{array}{l}n\\ x\end{array}\right)p^x{q}^{n-x}$

Where, n is the number of trials, x is the number of successes among n trials, p is the probability of successes and q is the probability of failures.