Question

Question 1 A researcher at a large university wanted to investigate if a student's seat preference was related in any way to the gender of the student. The researcher divided the lecture room into three sections (1-front, middle of the room, 2-front, sides of the classroom, and 3-back of the classroom, both middle and sides) and noted where his students sat on a particular day of the class. The researcher's summary table is provided below.

	Area (1)	Area (2)	Area (3)	Total
Males	17	9	7	33
Females	14	14	11	39
Total	31	23	18	72

(a) A person is randomly selected. Find the probability that the person is male or sits in the middle of the room. [3 marks]

(b) Suppose a person sitting in the front, middle portion of the class is randomly selected to answer a question. Find the probability the person selected is a female. [3 marks]

(c) Are the events ‘person is male’ and ‘sits in the front’ independent? Give reasons for your answer. [3 marks]

Answer 1

a) Let         
A:   Event that person is male        
B:   Event that the person sits in the middle of the room        
P(A) = Number of persons who are male / Total number of persons        
          = 33/72        
          = 0.4583        
P(B) = Number of persons who sit in the middle / Total number of persons        
          = 31/72        
           = 0.4306        
P(A∩B) = Number of persons who are male and sit in the middle / Total number of persons        
            = 17/72        
            = 0.2361        
Let         
C: Event that a person is male or sits in the middle of the room        
P(C ) = P(A) + P(B) - P(A∩B)        
           = 0.4583 + 0.4306 - 0.2361        
           = 0.6528        
P(person is male or sits in the middle of the room) = 0.6528        
         
b) Let         
D: Event that person is female        
E: Event that person is sitting in front, middle portion        
To find P(Person is female | person is sitting in front, middle portion)        
that is to find P(D | E)        
P(D | E) = P(D∩E) / P(E)        

      = 0.4516        
P(Person is female | person is sitting in front, middle portion) = 0.4516        
         
         
c) A:   Event that person is male        
B:   Event that the person sits in the front of the room        
P(A) = Number of persons who are male / Total number of persons        
          = 33/72        
          = 0.4583        
P(B) = Number of persons who sit in the front / Total number of persons        
          = (31+23)/72   (Front is Area1 + Area 2)     
           = 0.75        
P(A∩B) = Number of persons who are male and sit in the front / Total number of persons        
            = (17+9)/72        
            = 0.3611        
P(A) * P(B) = 0.4583 * 0.75        
   = 0.3437       
  ≠ P(A∩B)       
Hence, A and B are not independent        
Thus, the events ‘person is male’ and ‘sits in the front’ are NOT independent