Question 1019805
 
Question:
1. A small town has 5600 residents. The residents in the town were asked whether or not they favored building a new bridge across the river. You are given the following information on the residents' responses, broken down by gender:
                                    Men          Women          Total
          In Favor                  1400            280               1680
          Opposed                  840           3080               3920
                Total                2240           3360               5600
A) What is the probability of a randomly selected resident being in opposed to the bridge?
B) What is the probability that a randomly selected resident is a man and is Opposed to the bridge?
C) What is the probability of a randomly selected resident being a man or Opposed to the bridge?
D) If a randomly selected resident is a man, what is the probability that he is in favor of the bridge?
E) Are gender and opinion about the bridge mutually exclusive events? Why?
F) Are gender and opinion about the bridge independent events? Why? Show some "proof"
 
Solution:
 
The question gives a contingency table, from which we are expected to extract answers to the given questions.
We will define events:
O=opposition to project
~O=in favour of project
M=opinion by a male
~M=opinion by a female
We assume all samples are randomly selected, and size of sample is large (so that there is no repetition of selected subjects).
 
(a) 
P(O)=total number of oppositions / total number of subjects asked.
= 3920/5600
= 7/10
(b)
P(O∩M)
=840/5600
= 3/20
(c)
P(O∪M)
=P(O)+P(M)-P(M∩O)
=(3920+2240-840)/5600
= 19/20
(d)
P(~O|M)
=P(~O∩M)/P(M)
=(1400/5600)/(2240/5600)
=1400/2240
= 5/8
(e)
They are not mutually exclusive events, since every opinion is associated with a gender.
(f)
Gender and opinion are not independent events.
If they were independent, then P(M∩O)=P(M)*P(O)
Here P(M∩O)=840/5600=3/20
P(M)*P(O)=2240/5600*3920/5600=7/25
Since they are not equal, the two characteristics are not independent.