Eight identical slips of paper, each containing one number from one to eight, inclusive, are mixed up inside a bag. Subset A of the sample space represents the complement of the event in which the number 6 is drawn from the bag.
Which shows subset A?
A. [tex]$A={6}$[/tex]
B. [tex]$A={1,3,5,7}$[/tex]
C. [tex]$A={1,2,3,4,5,7,8}$[/tex]
D. [tex]$A={1,2,3,4,5,6,7,8}$[/tex]
Answer (2)
The sample space is the set of numbers from 1 to 8: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 .
The event of drawing a 6 is 6 .
The complement, subset A, includes all numbers in the sample space except 6.
Therefore, A = 1 , 2 , 3 , 4 , 5 , 7 , 8 .
Explanation
Understanding the Problem We are given a set of eight slips of paper, each containing a unique number from 1 to 8. We want to find the subset A, which represents the complement of the event where the number 6 is drawn. This means A contains all the numbers from 1 to 8, except for 6.
Identifying the Sample Space and the Event The sample space consists of all possible outcomes, which are the numbers from 1 to 8. So the sample space is {1, 2, 3, 4, 5, 6, 7, 8}. The event of drawing the number 6 is simply the set {6}. The complement of this event is the set of all elements in the sample space that are not in the event {6}.
Finding the Complement To find the complement, we remove the number 6 from the sample space. This gives us the set A = {1, 2, 3, 4, 5, 7, 8}.
Examples
In probability, understanding complements is crucial. For instance, if you're analyzing the likelihood of a certain event occurring (like drawing a specific number from a set), the complement helps you determine the probability of that event not occurring. This concept is widely used in risk assessment, statistical analysis, and decision-making processes where knowing the chances of an event not happening is as important as knowing the chances of it happening. For example, in a medical study, you might want to know the probability of a patient not experiencing side effects from a new drug. This is the complement of the event where a patient does experience side effects.
The complement of the event in which the number 6 is drawn includes all other numbers from 1 to 8 except 6. Therefore, subset A is represented as A = { 1 , 2 , 3 , 4 , 5 , 7 , 8 } , corresponding to option C. Thus, the correct answer is C.
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