In a population, 28 % of people have a disease called D, a certain kind of...

In a population, 28 % of people have a disease called D, a certain kind of skin problem, among them 42 % will develop into skin cancer. For people without this skin problem (D), only 8 % will have skin cancer.

a) What is the rate of skin cancer in this population

b) If a person found out that he has skin cancer, what is the chance that he had skin problem (D)?

Solutions

Expert Solution

Let A defines an event of getting a skin disease.

A^c defines an event of not getting skin disease.

It is given that 28% of people have D-disease. Hence, P(A) = 0.28 and P(A^c) = 0.72

Let B defines an event of developing skin cancer.

Among those diseased people 42% people will develop into skin cancer. Hence, probability that people will develop skin cancer given that they have skin disease D is 0.42. Hence, P( B|A ) = 0.42

For people without skin problem, only 8% will have skin cancer. Hence, probability of having skin cancer given that the person is not having skin disease is 0.08. That is, P( B|A^c ) = 0.08

(a) - We need to find the rate of skin cancer. That is P( B )

The Law of total probability states that,

Hence, the rate of skin cancer in the population is 17.52%

(b) -

We need to find the chance that the person had a skin problem given that he has a skin cancer. That is P( A|B )

According to the Baye's theorem,

Hence, If a person found out that he has a skin cancer, there is 67.12% chance that he had skin problem.

I hope you find the solution helpful. Feel free to ask any doubt in the comment section.

Please do not forget to vote the answer. Thank you in advance!!!

orchestra answered 1 year ago

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