Question

A 2010 Pew Research poll asked 1,306 Americans "From what you've read and heard, is there solid evidence that the average temperature on earth has been getting warmer over the past few decades, or not?". The table below shows the distribution of responses by party and ideology, where the counts have been replaced with relative frequencies.

	Earth is warming	Not warming	Don't know (or refuse)	Total
Conservative Republican	0.11	0.2	0.02	0.33
Mod/Lib Republican	0.06	0.06	0.01	0.13
Mod/Cons Democrat	0.25	0.07	0.02	0.34
Liberal Democrat	0.18	0.01	0.01	0.2
Total	0.6	0.34	0.06	1

b) What is the probability that a randomly chosen respondent believes the earth is warming or is a liberal Democrat?  (please round to four decimal places)

c) What is the probability that a randomly chosen respondent believes the earth is warming given that he is a liberal Democrat?  (please round to four decimal places)

d) What is the probability that a randomly chosen respondent believes the earth is warming given that he is a conservative Republican?  (please round to four decimal places)

f) What is the probability that a randomly chosen respondent is a moderate/liberal Republican given that he does not believe that the earth is warming?  (please round to four decimal places)

Answer 1

We will look for the derised probability from the table given as shown below:

b) The probability that a randomly chosen respondent believes the earth is warming or is a liberal Democrat.

Note: For event A & B, probability of A or B is given by  P(A U B) = P(A) + P(B) - P(A ∩ B).

So, P(earth is warming or liberal Democrat) = P(earth is warming) + P(liberal Democrat) - P(earth is warming and liberal Democrat)  

= 0.6 + 0.2 - 0.18

= 0.62

c) Probability that a randomly chosen respondent believes the earth is warming given that he is a liberal Democrat.

Note: Here we utilize law conditional probabilty i.e. Probability of event B given event A = P(B|A) = P(A∩B) / P(A)

So, P(earth is warming / given liberal Democrat) = 0.18 / 0.20 = 0.9

d) Probability that a randomly chosen respondent believes the earth is warming given that he is a conservative Republican.

Note: As above we utilize law of conditional probability.

So,  P(earth is warming / given conservative Republican) = 0.11 / 0.33 = 0.33

f) Probability that a randomly chosen respondent is a moderate/liberal Republican given that he does not believe that the earth is warming.

Note: As above we utilize law of conditional probability.

P(moderate/liberal Republican / given not warming) = 0.06/ 0.34 = 0.1765