SOLUTION: 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

Question 1019805: 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"
Found 2 solutions by ewatrrr, mathmate:
Answer by ewatrrr(24785) (Show Source): You can put this solution on YOUR website!

Hi
Nice Chart!
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? P = 3920/5600
B) What is the probability that a randomly selected resident is a man and is Opposed to the bridge? P = 840/5600
C) What is the probability of a randomly selected resident being a man or Opposed to the bridge?
P(A or B) = P(A) + P(B) - P(A and B)
P = (3920 - 840)/5600
D) If a randomly selected resident is a man, what is the probability that he is in favor of the bridge?
P(A|B) = P(A and B)/P(B) Bayes Theorem
P = (1400/5600) /(2240/5600) = 1400/2240
E) Are gender and opinion about the bridge mutually exclusive events? NO Why?
Mutually Exclusive: can't happen at the same time
F) Are gender and opinion about the bridge independent events? NO
Why? that A occurs does affect the probability of B occurring
independent: that A occurs does not affect the probability of B occurring.
Show some "proof"
% of men opposing (840/2240)*100=%men is less than (3080/3360)*100=%women
Answer by mathmate(429) (Show Source): You can put this solution on YOUR website!

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.

RELATED QUESTIONS
A small town has 5,636 residents. The residents in the town were asked whether or not... (answered by ikleyn)
A small town has 5,600 residents. The residents in the town were asked whether or not... (answered by stanbon)
A small town has 5,600 residents. The residents in the town were asked whether or not... (answered by stanbon)
The mayor of a town has proposed a plan for the annexation of an adjoining bridge. A... (answered by Boreal)
Town A and Town B were located close to each other, and recently merged into one city.... (answered by lwsshak3)
Town A and Town B were located close to each other, and recently merged into one city.... (answered by josmiceli)
Town A and Town B were located close to each other, and recently merged into one city.... (answered by ankor@dixie-net.com)
Town A and Town B were located close to each other, and recently merged into one city.... (answered by ikleyn,greenestamps)
A random sample of 722 residents in a major town was asked whether they had ever been... (answered by ewatrrr)