A probability experiment is conducted in which the sample space of the experiment is $S=\{1,2,3,4,5,6,7,8,9,10,11$, $12,13,14,15\}$. Let event $A=\{2,3,4,5\}$ and event $B=\{13,14,15\}$. Assume that each outcome is equally likely. List the outcomes in A and B. Are A and B mutually exclusive?

Select one:
a. $\{2,3,4,5,13,14,15\}$; yes
b. $\{\}$; no
c. $\{2,3,4,5,13,14,15\}$; no
d. $\{\}$; yes

Question

A probability experiment is conducted in which the sample space of the experiment is $S=\{1,2,3,4,5,6,7,8,9,10,11$, $12,13,14,15\}$. Let event $A=\{2,3,4,5\}$ and event $B=\{13,14,15\}$. Assume that each outcome is equally likely. List the outcomes in A and B. Are A and B mutually exclusive? 
 
Select one: 
a. $\{2,3,4,5,13,14,15\}$; yes 
b. $\{\}$; no 
c. $\{2,3,4,5,13,14,15\}$; no 
d. $\{\}$; yes

GinnyAnswer · Answer

The problem involves determining if two events, A and B, are mutually exclusive given their outcomes. Since the intersection of A and B is an empty set, A and B are mutually exclusive. The outcomes in A are { 2 , 3 , 4 , 5 } and the outcomes in B are { 13 , 14 , 15 } . A ∩ B = ∅ . Therefore, A and B are mutually exclusive. 
 Explanation 
 
 Problem Analysis The sample space is S = { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 } . Event A = { 2 , 3 , 4 , 5 } and event B = { 13 , 14 , 15 } . We want to list the outcomes in A and B and determine if A and B are mutually exclusive. 
 
 List Outcomes The outcomes in A are 2 , 3 , 4 , 5 . The outcomes in B are 13 , 14 , 15 . 
 
 Find Intersection To determine if A and B are mutually exclusive, we need to find their intersection, A ∩ B . The intersection of two sets is the set of elements that are common to both sets. In this case, A = { 2 , 3 , 4 , 5 } and B = { 13 , 14 , 15 } . There are no elements in common between A and B. Therefore, A ∩ B = ∅ , which is the empty set. 
 
 Determine Mutual Exclusivity Since the intersection of A and B is the empty set, A and B are mutually exclusive. 
 
 Final Answer The outcomes in A and B are { 2 , 3 , 4 , 5 } and { 13 , 14 , 15 } , respectively. Since A ∩ B = ∅ , A and B are mutually exclusive. Therefore, the correct answer is (d).

Examples 
 In probability, mutually exclusive events are important in many real-world scenarios. For example, consider a medical diagnosis: a patient cannot simultaneously have two mutually exclusive diseases. Similarly, in a game of chance, certain outcomes might be mutually exclusive, such as rolling a 1 or a 6 on a single die. Understanding mutually exclusive events helps in calculating probabilities accurately in these situations.

Answer (1)

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