Problem

A probability experiment is conducted in which the sample space of the experiment is $S=\{9,10,11,12,13,14,15,16,17,18,19,20\}$. Let event $E=\{12,13,14,15,16,17,18,19,20\}$. Assume each outcome is equally likely. List the outcomes in $E^{C}$. Find $P\left(E^{C}\right)$.
List the outcomes in $E^{\mathrm{C}}$. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. $E^{c}=\{9,10,11\}$
(Use a comma to separate answers as needed.)
B. $E^{C}=\{\}$
$P\left(E^{C}\right)=\square$ (Type an integer or a decimal rounded to three decimal places as needed.)

Answer

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Answer

So, the outcomes in $E^{C}$ are $\boxed{\{9,10,11\}}$ and $P\left(E^{C}\right)=\boxed{0.25}$.

Steps

Step 1 :The sample space of the experiment is $S=\{9,10,11,12,13,14,15,16,17,18,19,20\}$ and event $E=\{12,13,14,15,16,17,18,19,20\}$.

Step 2 :The complement of an event E, denoted by $E^C$, is the set of all outcomes in the sample space that are not in E.

Step 3 :So, we need to find the elements in the sample space S that are not in E. The outcomes in $E^{C}$ are $\{9,10,11\}$.

Step 4 :We can calculate the probability of $E^C$ by dividing the number of outcomes in $E^C$ by the total number of outcomes in the sample space S. Since each outcome is equally likely, this will give us the correct probability.

Step 5 :The probability of $E^C$ is $\frac{3}{12}=0.25$.

Step 6 :So, the outcomes in $E^{C}$ are $\boxed{\{9,10,11\}}$ and $P\left(E^{C}\right)=\boxed{0.25}$.

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