Probability lung cancer (statistics)

Question

Approximately 1 percent of Americans ages 50-60 have lung cancer. A patient with lung cancer has a 87 percent chance of a positive test. A patient without lung cancer has a 9 percent chance of a positive test (a false positive result).

Complete the table below, then answer the questions below the table. Give all answers as decimals to 4 decimal places.

	Positive test result	Negative test result	Total
Has lung cancer
Does not have lung cancer
Total

What is the probability a patient has lung cancer given that she just had a positive test?

Give answer as decimals to 4 decimal places.

Answer 1

(1) We know that the since all the marginal probabilities must sum to 1 that:

P(Has Cancer)+P(Without Cancer)=1

P(Without Cancer)=1-P(Has Cancer)

P(Without Cancer)=1-.01

P(Without Cancer)=.99

(2) Excellent, now we must consider P(+|Has Cancer)=.87. We know that because this is a conditional probability, the formula must be: P(+|Has Cancer)=P(+ ∩ Has Cancer)/P(Has Cancer). Thus, we know

P(+ ∩ Has Cancer)/P(Has Cancer)=.87

We know the probability of having cancer is .01, so all we have to do is plug it in:

P(+ ∩ Has Cancer)=P(Has Cancer)×.87

P(+ ∩ Has Cancer)=.01×.87

P(+ ∩ Has Cancer)=.0087

(3) We also know that the since all the probabilities must sum up to the marginal probabilities:

P(+ ∩ Has Cancer) + P (- ∩ Has Cancer) = .0100

P (- ∩ Has Cancer) = .0100 – P(+ ∩ Has Cancer)

P (- ∩ Has Cancer) = .0100 – .0087

P (- ∩ Has Cancer) = .0013

(4) Now, let's use the final piece of information given: P(+|Without Cancer)=.09.

We know that because this is a conditional probability, the formula must be: P(+|Without Cancer)=P(+ ∩ Without Cancer)/P(Without Cancer). Thus, we know

P(+ ∩ Without Cancer)/P(Without Cancer)=.09

We know the probability of being without cancer is .99, so all we have to do is plug it in:

P(+ ∩ Without Cancer)=P(Without Cancer)×.09

P(+ ∩ Has Cancer)=.99×.09

P(+ ∩ Has Cancer)=.0891

(5) We also know that the since all the probabilities must sum up to the marginal probabilities, therefore:

P(+) = P(+ ∩ Has Cancer) + P (+ ∩ Without Cancer)

P(+) = .0087 + .0891

P(+) = .0978

(6) We also know that the since all the marginal probabilities must sum to 1 that:

P(+) + P(-) = 1

P(-) = 1 - P(+)

P(-) = 1 - .0978

P(Without Cancer)=.9022

(7) We also know that the since all the probabilities must sum up to the marginal probabilities, therefore:

P(Without Cancer) = P(+ ∩ Without Cancer) + P (- ∩ Without Cancer)

P (- ∩ Without Cancer) = P(Without Cancer) – P(+ ∩ Without Cancer)

P (- ∩ Without Cancer) = .9900 – .0891

P(+) = .9009

Excellent, with this in mind, let's fill it in:

	Positive test result	Negative test result	Total
Has lung cancer	.0087	.0013	.0100
Does not have lung cancer	.0891	.9009	.9900
Total	.0978	.9022	1.0000

We have filled in the whole box rounded to 4 digits!