Question

In a survey of 233 people, the following data were obtained relating gender to political orientation:

	Republican (R)	Democrat (D)	Libertarian (L)	Total
Male (M)	53	55	7	115
Female (F)	62	44	12	118
Total	115	99	19	233

A person is randomly selected. What is the probability that the person is:
a) Male?
b) Male and a Democrat?
c) Male given that the person is a Democrat?
d) Republican given that the person is Male?
e) Female given that the person is a Libertarian?
f) Are the events Male and Republican independent? yes or no .

Answer 1

Solution:-

We have given data of N=233 people as follows.

	Republican	Democrat	Libertarian	Total
Male (M)	53	55	7	115
Female (F)	62	44	12	118
Total	115	98	19	N=233

Question)If a person is randomly selected, what is Probability that

a)Male?

---->Probability (Male) =( total no. Of male)/N

Probability (Male) = 115/233

b)male and a democrat?

------>

Probability (Male and democrat)

=( Total no.of male who are democrat)/N

=55/233

c)Male given that the person is a Democrat?

-----> that is,

Probability (Male given that the person is a Democrat)

=Probability(male & democrat)/Probability (Democrat)

...........{ Definition.... P(A given B) = P(A & B )/P(B) }

But,

Probability (Democrat) =( Total person in democrat)/N

=99/233

=0.424893........(i)

Therefore from (i) and "b"

Probability (Male given that the person is a Democrat)

=0.236051/0.424893

d) Republican given that the person is male?

------>That is,

Probability (Republican given that the person is male)

=Probability (Republican & male)/Probability (male)

But,

Probability (Republican & male)

=( Total Republican person which are male)/N

=53/233

=0.227467.........(ii)

Therefore, from"a" and (ii)

Probability (Republican given that the person is male)

=0.227467/0.493562

=0.460869

e)Female given that the person is a Libertarian?

-----> that is,

Probability (Female given that person is a Libertarian)

=Probability(Female & libertarian)/Probability (libertarian)

But

Probability (female & libertarian)

= (Total no.of female who are libertarian)/N

=12/233

=0.051502.........(iii)

And

Probability (libertarian) = (Total no.of libertarian)/N

= 19/233

=0.081545........(iv)

Therefore from (iii) and ( iv)

Probability (Female given that person is a Libertarian)

=0.051502/0.081545

=0.631579

f)Are the events Male and Republican independent?

-----> by definition of independence,

If two events A and B are independent if and only if

P(A given B) = P(A)

Consider,

Probability ( Republican)

=(Total no.of Republicans)/N

=115/233

=0.493562............(v)

And

From "d"

Probability (Republican given male) = 0.460869......(vi)

From (v) and (vi)

Probability (Republican given male) is not equal to probability (Republican)

Therefore,