Question

A test for ovarian cancer has a 4 percent rate of false positives and a 1 percent rate of false negatives. On average, 1 in every 2,000 American women over age 35 actually has ovarian cancer. If a woman over 35 tests positive, what is the probability that she actually has cancer

Answer 1

Answer:

The probability that a woman over 35 actually has cancer given that she tests positive is 0.012.

Step-by-step explanation:

The information provided is:

P (+ | X') = 0.04

P (- | X) = 0.01

P (X) = 0.0005

Compute the value of P (+ | X) as follows:

P (+ | X) = 1 - P (- | X)

            = 1 - 0.01

            = 0.99

Compute the value of P (+) as follows:

P (+) = P (+ | X) × P (X) + P (+ | X') × P (X')

       $=(0.99\times 0.0005)+(0.04\times (1-0.0005))\\\\=0.000495+0.03998\\\\=0.040475\\\\\approx 0.0405$

Compute the probability that a woman over 35 actually has cancer given that she tests positive as follows:

$P(X|+)=\frac{P(+|X)P(X)}{P(+)}$

             $=\frac{0.99\times 0.0005}{0.0405}\\\\=0.0122222\\\\\approx 0.012$

Thus, the probability that a woman over 35 actually has cancer given that she tests positive is 0.012.