Statistics Question 2?

Question

A large research hospital has accumulated statistical data on its patients for an extended period. Researchers have determined that patients who are smokers have an 18% chance of contracting a serious illness, whereas there is only a 0.06 probability that a nonsmoker will contract a serious illness. From hospital records, the researchers know that 23% of all hospital patients are smokers, while 77% are nonsmokers. For planning purposes, the hospital physician staff would like to know the probability that a given patient is a smoker if the patient has a serious illness.

a) 0.04

b) 0.09

c) 0.47

d) 0.58

Answer 1

This is an exercise in conditional probability.  You can use Bayes' Rule to solve this.  However, I find that it is much easier to just use the definition of conditional probability.  In short I derived Bayes' Rule every time I do this but it prevents me from making errors in more complex problems.

This is an exercise in conditional probability.  For any two events A and B, where P(B) ≠ 0, you have the conditional probability:

P( A | B ) = P( A ∩ B ) / P( B ) = P( B | A) * P(A) / P(B)

the above is read as: the probability of A given B is equal to the probability of A and B divided by the probability of B.

Use The Law of Total Probability to find P(B)

For a set of events A1, A2, A3, ... , An where the Ai's are mutually exclusive and exhaustive events and for any other event B

P(B)

= P(B and A1) + P(B and A2) + ... + P(B and An)

= P(B | A1) * P(A1) + P(B | A2) * P(A2) + ... + P(B | An) * P(An)

Let D be the event of having a serious illness

Let S be the event of being a smoker

Let ~S be the event of not being a smoker

we know

P( D | S ) = 0.18

P( D | ~S) = 0.06

P(S) = 0.23

P(~S) = 0.77

Find P( S | D )

P( S | D ) = P( D | S ) * P( S ) / P( D )

P(D) = P(D | S) * P(S) + P(D | ~S) * P(~S)

= 0.18 * 0.23 + 0.06 * 0.77

= 0.0876

P( S | D ) = P( D | S ) * P( S ) / P( D )

= 0.18 * 0.23 / 0.0876

= 0.4726027

the solution is (C)

Answer 2

a is the answer.  .18 *.23 = .041

Answer 3

The answer is A - .04.  Since 23 % will be smokers and of those 18% have a chance of serious illness, multiply .23 x .18 = .04 or 4%