Question

Kevin’s family has a history of diabetes. The probability that Kevin would inherit this disease is 0.75. Kevin decides to take a test to check if he has the disease. The accuracy of this test is 0.85.

We now know that the test predicted that Kevin does not have the disease. Using this information, calculate the probability that Kevin does not have diabetes after the test was taken.

0.0925
0.3565
0.4827
0.6538
0.8765

Answer 1

@YanaSidlinskiy

Answer 2

if A and B are two independent events, then
P(A∩B) = P(A)xP(B)
The probability that Kevin inherits the disease and the accuracy of the test are independent, so the intersection of the two probabilities = **** ?

Answer 3

let A=kevin has the disease
let B=the test is positive for the disease
We want the probability of A given B, or\[P(A|B)=\frac{P(A\cap B)}{P(B)}\]

Answer 4

Are we going to be drawing a tree? :)

Answer 5

the events are not independent - having a positive test changes the likelihood of having the disease and vice versa

Answer 6

yes, tree is a good idea

Answer 7

May you draw it :)

Answer 8

let's draw it together
the first fork should be between having the disease and not having the disease. What does that look like?

Answer 9

.85 and .15

Answer 10

that is the probability of the test being accurate
what is the probability of having the disease?
what is the probability of not having the disease?

Answer 11

probability of him having it is .75 not having it is .25

Answer 12

right, so we get a figure like sonow, if he has the disease (A), what is the probability that the test is positive (B) ?

Answer 13

Oh, I thought they were independent.

Answer 14

.15?

Answer 15

To decide if two events are independent, just think about whether or not knowing one piece of information influences your belief about the other. Doesn't knowing the test is positive influence your belief about whether or not he has the disease?

Answer 16

@cole1117 so you claim that if I have the disease, the test will only correctly tell me that I have the disease 15% of the time? Why?

Answer 17

85% accuracy means that if you have the disease, the test should be positive 85% of the time, and if you don't have it, the test should be negative 85% of the time.

Answer 18

I don not know. I thought that was what the question stated he only had a 15% chance of the test being incorrect.

Answer 19

85% accuracy means that if you have the disease, the test should be positive 85% of the time, and if you don't have it, the test should be negative 85% of the time.

If he has the disease (A) what is the probability that the test is positive (B)?

Answer 20

85

Answer 21

right, and what is the probability that the test is negative?

Answer 22

15

Answer 23

Yes, so that will look like so

Answer 24

i had mislabeled the bottom branch before, oops

Answer 25

It is fine. Now what?

Answer 26

what is the probability of the test being positive when he doesn't have the disease?

Answer 27

85

Answer 28

no no 15

Answer 29

right, good save

Answer 30

Yea I wasn't thinking at first.

Answer 31

ok, so now to our formula
we know the test is positive, and given that we want the probability of him having the disease
this is\[P(A|B)=\frac{P(A\cap B)}{P(B)}\]how are we still?

Answer 32

what does mean?