Question

Tick bites can lead to a red meat allery called the alpha-gal syndrome. Suppose that your friend, who loves hiking in the woods, has been experiencing odd medical symptoms lately. She is tested for alpha-gal syndrome. Let A be the event that your friend has alpha-gal syndrome, and suppose P{A} = 0.02. The allergy test is imperfect and many not correctly detect alpha-gal syndrome. Let D be the event that the test detects that the disease is present, and suppose P{D|A} = 0.97 and P{D^c|A^c} = 0.94.

What is the probability that the allergy test detects that the disease is present in your friend?

Answer 1

Solution:

Given:

A =  the event that your friend has alpha-gal syndrome and P(A) = 0.02

Thus P(A^c ) = 1 - P(A) = 1 - 0.02= 0.98

D = the event that the test detects that the disease is present

P(D|A) = 0.97

and P(D^c |A^c ) = 0.94

and we have to find:

P(D) =...................?

We can find answer in two ways:

First way:

P(D) = P(D|A) * P(A) + P( D | A^c) * P(A^c)

Since P(D^c |A^c ) = 0.94

then P(D |Ac) = 1- P(D^c |A^c ) = 1- 0.94 = 0.06

Thus

P(D) = P(D|A) * P(A) + P( D | A^c) * P(A^c)

P(D) = 0.97 * 0.02 + 0.06 * 0.98

P(D) = 0.0194 + 0.0588

P(D) = 0.0782

Second Way:

We have:

P(D|A) = 0.97

then using conditional probability formula:

and

Now using De'Morgans Law:

( Using addition rule of probability)

Thus the probability that the allergy test detects that the disease is present in your friend is: