Question

Based on recent statistics, the probability that a smoker who lives in a rural area will die from lung cancer is 0.00065. The probability that a smoker from an urban area will die from lung cancer is 0.00085. The probability that a nonsmoker from an urban area will die from lung cancer is 0.00015, whereas the probability that a rural nonsmoker will die from lung cancer is 0.00001. Approximately 70 percent of the population live in urban areas, and about 30 percent live in rural areas. Assume that 20 percent of urban dwellers are smokers and that 10 percent of rural d0wellers are smokers.

Given only the information that a person died from lung cancer, what is the probability that this person was an urban dweller?

Question 8 options:

	0.6
	0.74
	0.82
	0.9

Answer 1

We are given here that:
P( die from lung cancer | rural smoker ) = 0.00065,
P( die from lung cancer | urban smoker ) = 0.00085,
P( die from lung cancer | urban non smoker ) = 0.00015,
P( die from lung cancer | rural non smoker ) = 0.00001,

Also, we are given here that:
P( urban ) = 0.7, P(rural ) = 0.3
Also, we are given here that:
P( smoker | urban) = 0.2,
P( smoker | rural) = 0.1

Therefore,
P( urban smoker) =P( smoker | urban)P( urban ) = 0.2*0.7 = 0.14,
Therefore P( urban non smoker) = P(urban) - P( urban smoker )
Therefore P( urban non smoker) = 0.7 - 0.14 = 0.56

P( rural smoker ) = P( smoker | rural)P(rural ) = 0.1*0.3 = 0.03,
Therefore P( rural non smoker) = P(rural) - P( rural smoker )
Therefore P( rural non smoker) = 0.3 - 0.03 = 0.27

Therefore using law of total probability, we get here:
P( die from lung cancer)
= P( die from lung cancer | rural smoker ) P( rural smoker ) +
P( die from lung cancer | urban smoker ) P( urban smoker ) +
P( die from lung cancer | rural non smoker ) P( rural non smoker ) + P( die from lung cancer | urban non smoker ) P( urban non smoker )
= 0.00065*0.03 + 0.00085*0.14 + 0.00015*0.56 + 0.00001*0.27

= 0.0002252

Using Bayes theorem, we get here:
P( urban | die from lung cancer)

= [ P( die from lung cancer | urban smoker ) P( urban smoker ) + P( die from lung cancer | urban non smoker ) P( urban non smoker ) ] / P( die from lung cancer)

= (0.00085*0.14 + 0.00001*0.27 ) / 0.0002252

= 0.6

Therefore 0.6 is the required probability here.