Question

SUppose that a test for a disease correctly gives positive results of 95% of those having the disease and correctly gives negative results for 90% of those who don't have the disease. Suppose also that the incidence of the disease is 1%. If a person tests positive for the disease, what is the chance that they have the disease?

Answer 1

not as large as you might think
bayes formula, right?

Answer 2

bayes formula is kind of a pain to keep track what is what, but there is a somewhat easier way to solve this problem

Answer 3

has to do with true positives and true negatives i think

Answer 4

imagine that \(1000\) people are tested and that the percentages are exact
how many will have the disease?

Answer 5

95% of 1000?

Answer 6

hold on lets go slow a bit
and lets make the number even bigger to get nice whole number answers
imagine that you test 100, 000 people
what percent of the population has the disease?

Answer 7

uhh

Answer 8

forget about true positives etc,
you are told what percent of the population has the disease

Answer 9

in this line
"Suppose also that the incidence of the disease is 1%"

Answer 10

one percent of the population has the disease
you test 100,000 people, how many will actually have the disease?

Answer 11

1000?

Answer 12

yes one percent of 100,000 is 1,000

Answer 13

now we go to this line
Suppose that a test for a disease correctly gives positive results of 95%
of those 1,000 how have it, how many will test positive?

Answer 14

*who have it

Answer 15

950...

Answer 16

don't u do 95% of 1,000

Answer 17

good
now lets hold on to that number and continue

Answer 18

ok

Answer 19

we started with 100,000 people, and 1,000 have the disease
how many do NOT have the disease?

Answer 20

99000?

Answer 21

yes

Answer 22

now to this line
(the test) correctly gives negative results for 90% of those who don't have the disease
of the 99,000 who do not have the disease, how many negative, and how many test positive?

Answer 23

thats where i get confused.

Answer 24

ok what i am asking is what is 90% of 99,000 and what is 10% of 99,000

Answer 25

actually we really only need 10% of 99,000 for this problem, but if we know one number we know the other

Answer 26

90% = 89100 10%= 9900

Answer 27

ok so far so good

now we can answer the question
we know that \(9900\) of the people who do NOT have the disease test POSITIVE
and also that \(950\) of the people who DO have the disease also test POSITIVE

how many people total test positive ?

Answer 28

10850?

Answer 29

that is what i get as well

Answer 30

and now for the answer:
of those \(10850\) who test positive, how many actually HAVE the disease?

Answer 31

950?

Answer 32

yes
so the probability that if you test positive, you actually have the disease is
\[\frac{950}{10850}\]

Answer 33

a fraction that you can certainly reduce, but that is the answer

Answer 34

My choices for an answer are 8.8%, 10%, 13.4% 95%

Answer 35

use a calculator to make it in to a percent

Answer 36

the percent chance that they have the disease :3

Answer 37

or estimate, since it is certainly under ten percent

Answer 38

8.8

Answer 39

ok that looks reasonable

Answer 40

Could you help me with 2 more for tonight! I have given you medals I believe and you are amazingly helpful!

Answer 41

we can redo this problem without the 100,000 if you like but it is often easier to do it with numbers than with percents

Answer 42

the numerator was \(.95\times .01=.0095\) and the denominator was
\[.95\times .01+.1\times .99\]

Answer 43

in this case i would go with A just because it is the only answer that makes sense