Scenario 1: You go to see the doctor about an ingrown toenail. The doctor select

Question

Scenario 1: You go to see the doctor about an ingrown toenail. The doctor selects you at random to have a blood test for swine flu, which for the purposes of this exercise we will say is currently suspected to affected 1 in 10,000 people in the U.S. The test has a sensitivity of 100% and a specificity of 99%. Scenario 2: Given the same sensitivity and specificity of the swine flu test from above, imagine that you went to a friend’s wedding in Mexico recently. It is known (for the purposes of this exercise) that 1 in 200 people who visited Mexico recently come back with swine flu. Upon your return your doctor requires you have a blood test.

6) In both scenarios, you test positive.

a) What is the probability that you have swine flu after your trip to the doctor for an ingrown toenail? Answer with a percent to the nearest whole number.

b) What is the probability that you have swine flu after your trip to Mexico? Answer with a percent to the nearest whole number.

Explanation / Answer

ans=

Let P(D) be the probability you have swine flu.

Let P(T) be the probability of a positive test.

We wish to know P(D|T).

Bayes theorem says

P(D|T) = P(T|D)P(D) / P(T)

which in this case can be rewritten as

P(D|T) = P(T|D)P(D) / (P(T|D)P(D) + P(T|ND)P(ND))

where P(ND) means the probability of not having swine flu.

We have

P(D) = 0.0001 (the a priori probability you have swine flu).

P(ND) = 0.9999

P(T|D) = 1 (if you have swine flu the test is always positive).

P(T|ND) = 0.01 (1% chance of a false positive).

Plugging these numbers in we get

P(D|T) = 1*0.0001 / (1*0.0001+0.01*0.9999) 0.01

That is, even though the test was positive your chance of having swine flu is only 1%.

However, if you went to Mexico recently then your starting P(D) is 0.005. In this case

P(D|T) = 1*0.005 / (1*0.005+0.01*0.995) 0.33

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