A new test has been devised for detecting a particular type of cancer. If the test is applied to a person who has this type of cancer, the probability that the person will have a positive reaction is 0.95 and the probability that the person will have a negative reaction is 0.05. If the test is applied to a person who does not have this type of cancer, the probability that the person will have a positive reaction is 0.05 and the probability that the person will have a negative reaction is 0.95. Suppose that in the general population, one person out of every 100,000 people has this type of cancer. If a person selected at random has a positive reaction to the test, what is the probability that he has this type of cancer?

Question

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Respuesta :

valetta valetta · Answer 1 · 2019-10-10T08:02:55+02:00

Answer:

0.000189

Step-by-step explanation:

Given:

Probability of positive reaction given that he has cancer, P(p/c) = 0.95

Probability of negative reaction given that he has cancer, P(n/c) = 0.05

Probability of positive reaction given that not having cancer, P(p/c') = 0.05

Probability of negative reaction given that not having cancer, P(n/c') = 0.95

Probability that the person has particular type of cancer,

P(C) = [tex]\frac{\textup{1}}{\textup{100,000}}=0.00001[/tex]

Probability that the person does not has particular type of cancer,

P(C') = 1 - 0.00001 = 0.99999

Now,

Using the Baye's Theorem

Probability that the person selected at random has a positive reaction to the test, that he has this type of cancer

= [tex]\frac{P(p/c)\times P(C)}{P(p/c)\times P(C)+P(p/c')\times P(C')}[/tex]

= [tex]\frac{0.95\times0.00001}{0.95\times0.00001+0.05\times0.99999}[/tex]

= 0.000189