Question

a probability experiment is conducted in which the sample space of the experiment is $s = \\{1,2,3,4,5,6,7,8,9,10,11,12\\}$. let event $e = \\{1,2,3\\}$. assume each outcome is equally likely. list the outcomes in $e^{c}$. find $p(e^{c})$. list the outcomes in $e^{c}$. select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. $e^{c} = \square$ (use a comma to separate answers as needed.) b. $e^{c} = \\{\\}$ $p(e^{c}) = \square$ (type an integer or a decimal rounded to three decimal places as needed.)

Explanation:

Step1: Find the complement of event \(E\)

The complement of event \(E\), denoted as \(E^{c}\), consists of all outcomes in the sample space \(S\) that are not in \(E\).
Given \(S = \{1,2,3,4,5,6,7,8,9,10,11,12\}\) and \(E=\{1,2,3\}\), then \(E^{c}=\{4,5,6,7,8,9,10,11,12\}\)

Step2: Calculate the probability of \(E^{c}\)

The probability formula is \(P(A)=\frac{n(A)}{n(S)}\), where \(n(A)\) is the number of elements in event \(A\) and \(n(S)\) is the number of elements in the sample space.
Here, \(n(E^{c}) = 9\) and \(n(S)=12\)
So \(P(E^{c})=\frac{9}{12}=\frac{3}{4}= 0.75\)

Answer:

A. \(E^{c}=\{4,5,6,7,8,9,10,11,12\}\)
\(P(E^{c}) = 0.75\)