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 $E=\{2,3,4,5,6,7\}$ and event $F=\{6,7,8,9\}$. List the outcomes in $E$ and F. Are $\mathrm{E}$ and $\mathrm{F}$ mutually exclusive?
List the outcomes in $\mathrm{E}$ and $\mathrm{F}$. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. $\{6,7\}$ (Use a comma to separate answers as needed.)
B. \{\}
Are $\mathrm{E}$ and $\mathrm{F}$ mutually exclusive?
A. Yes. $\mathrm{E}$ and $\mathrm{F}$ have outcomes in common.
B. No. $\mathrm{E}$ and $\mathrm{F}$ have no outcomes in common.
C. Yes. $\mathrm{E}$ and $\mathrm{F}$ have no outcomes in common.
D. No. $\mathrm{E}$ and $\mathrm{F}$ have outcomes in common.
Answer
Final Answer: The outcomes in E and F are \(\boxed{\{6,7\}}\). E and F are not mutually exclusive, so the answer is \(\boxed{\text{D. No. E and F have outcomes in common.}}\)
Step 1 ：The outcomes in events E and F are given by E = \{2, 3, 4, 5, 6, 7\} and F = \{8, 9, 6, 7\}.
Step 2 ：The common outcomes in events E and F can be found by finding the intersection of the two sets, which gives \{6,7\}.
Step 3 ：Two events are said to be mutually exclusive if they cannot occur at the same time, i.e., the intersection of the events is an empty set.
Step 4 ：Checking the intersection of E and F, we find that it is not an empty set.
Step 5 ：Therefore, E and F are not mutually exclusive because they have outcomes in common.
Step 6 ：Final Answer: The outcomes in E and F are \(\boxed{\{6,7\}}\). E and F are not mutually exclusive, so the answer is \(\boxed{\text{D. No. E and F have outcomes in common.}}\)
