Question

5. Two fair, distinct dice (one red and one green) are rolled. Let A be the event the red die comes up even and B be the event the sum on the two dice is 8. Are A,B independent events?

According to the American Lung Association 7% of the population has lung disease. Of the people having lung disease 90% are smokers. Of the people not having lung disease 20% are smokers. What are the chances that a smoker has lung disease?

Answer 1

Question 1:

The various combinations for the 2 dice throws here are computed as:

	Red = 1	Red = 2	Red = 3	Red = 4	Red = 5	Red = 6
Green = 1		A		A		A
Green = 2		A		A		A,B
Green = 3		A		A	B	A
Green = 4		A		A,B		A
Green = 5		A	B	A		A
Green = 6		A,B		A		A

we have here:

P(A and B) = 3/36 = 1/12
P(A) = 18/36 = 1/2
P(B) = 5/36
P(A)P(B) = 5/72 which is not equal to P(A and B)

Therefore A and B are not independent here.

Question 2:

Here, we are given that:
P( lung disease ) = 0.07 , therefore P( no lung disease ) = 0.93
P( smokers | lung disease ) = 0.9, therefore P(non smoker | lung disease ) = 0.1
P( smokers | no lung disease ) = 0.2, therefore P(non smoker | no lung disease ) = 0.8

Using law of total probability, we get here:
P( smokers) = P( smokers | lung disease )P( lung disease ) + P( smokers | no lung disease ) P(no lung disease )

P(smokers) = 0.9*0.07 + 0.2*0.93 = 0.249

Using bayes theorem, we get here:
P( lung disease | smokers) = P( smokers | lung disease )P( lung disease )/P( smokers)

P( lung disease | smokers) = 0.9*0.07/0.249 = 0.2530

Therefore 0.2530 is the required probability here.