According to the American Lung Association, there is a 0.13% chance to develop l

Question

According to the American Lung Association, there is a 0.13% chance to develop lung cancer. Of the people who have lung cancer. 90% of them are smokers. In the population of people who do not have lung cancer, 16.9% are smokers. (a) If you are a smoker, what is your probability to develop lung cancer? (b) If you are not a smoker, what is your probability to develop lung cancer? You have a bucket of 10 red balls and 10 green balls. I remove one ball from the bucket. I make a note of the color of the ball, but do not tell you. Next, I replace the ball along with 10 new balls with the same color as the ball that I selected. After all of this, you select a ball from the bucket. (a) Draw a probability tree diagram, where the first level is the color I select, and the second level is the color that you select. Be sure to give all branches of the tree and their probabilities, as well as the joint probabilities at the end of the branches! (b) Using your diagram, what is the probability that the ball you selected is red? (c) Consider the two events A = "I pick green" and B = "you pick red". Are these events independent? (Don't just answer yes or no, show your work!) (d) If you pick a red ball, what is the probability that I picked a red ball?

Explanation / Answer

Question 2:

Here we are given the probability to get lung cancer is 0.13%. Therefore,

P( Cancer) = 0.0013, therefpre P( no cancer ) = 1- 0.0013 = 0.9987

Also we are given that the prople who have lung cancer, 90% are smokers, therefore

P( Smoker | Cancer ) = 0.9 therefore P( non smoker | Cancer) = 1-0.9 = 0.1

Also 16.9% of the population without cancer is a smoker. Therefore,

P( Smoker | No Cancer) = 0.169

a) Using addition law of probability we have:

P( Smoker ) = P( Smoker | Cancer )P(Cancer ) + P( Smoker | No Cancer)P( No cancer)

P( Smoker )= 0.9*0.0013 + 0.169*0.9987 = 0.1699503

Now by Bayes theorem we get:

P( Cancer | Smoker ) P( Smoker ) = P( Smoker | Cancer ) P( cancer )

Putting all the values we get:

P( Cancer | Smoker ) *0.1699503 = 0.9*0.0013

Therefore, we get: P( Cancer | Smoker ) = 0.9*0.0013 /0.1699503 = 0.0069

Therefore 0.0069 is the required probability here.

b) Now we know that P( smoker ) = 0.1699503, therefore P( no smoker ) = 1 - 0.1699503 = 0.8300497

Now by bayes theorem we get:

P(Cancer | no smoker )P( no smoker ) = P( non smoker | Cancer) P(cancer)

Putting all the values we get:

P(Cancer | no smoker )*0.8300497 = 0.1*0.0013

P(Cancer | no smoker ) = 0.1*0.0013/0.8300497 = 0.000157

Therefore 0.000157 is the required probability here.

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