Question

# A​ new, simple test has been developed to detect a particular type of cancer. The test...

A​ new, simple test has been developed to detect a particular type of cancer. The test must be evaluated before it is put into use. A medical researcher selects a random sample of 2 comma 000 adults and finds​ (by other​ means) that 2​% have this type of cancer. Each of the 2 comma 000 adults is given the​ test, and it is found that the test indicates cancer in 99​% of those who have it and in 1​% of those who do not. Based on these​ results, what is the probability of a randomly chosen person having cancer given that the test indicates​ cancer? Of a person having cancer given that the test does not indicate​ cancer?

Solution :

Given that,

P(no cancer positive) = 0.99 * 0.01 = 0.0099

P( cancer positive) = 0.02 * 0.99 = 0.0198

The probability that the positive test = 0.0099 + 0.0198 = 0.0297

(a)

the probability of a randomly chosen person having cancer given that the test indicates​ cancer is

= 0.0099 / 0.0297

= 0.33

Probability = 0.33

(b)

The probability of a person without cancer having a negative test is ,

= 0.99 * 0.99 = 0.9801

The probability of a person with cancer having a negative test is,

= 0.02 * 0.02 = 0.0004

A person having cancer given that the test does not indicate​ cancer is,

= 0.004 / 0.0004 + 0.9801

= 0.0004 / 0.9805

= 0.0004

Probability = 0.0004

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