Question

I need someone to explain this to me, I really don't understand...
Using Baye's Theorem:
It is known that 2% of the population has a certain allergy. A test correctly identifies people who have the allergy 98% of the time. The test correctly identifies people who do not have the allergy 94% of the time. A doctor decides that anyone who tests positive for the allergy should begin taking anti-allergy medication. Do you think this is a good decision? Why or why not?

I know it's pretty long and a little bit of reading but it's on paper so I had to copy it, I wouldn't have if I didn't need the

Answer 1

this is often very confusing
one good way to approach this is to use some actual numbers and see what happens

Answer 2

it makes "baye's theorem" far more undersantable

Answer 3

Okay but how do I choose my numbers?

Answer 4

i would pick a large number for the population, so that we get all whole number answers (although once we get the answer, we can redo it without using that crutch)
lets say the population is 1000 people ok?

Answer 5

okay

Answer 6

and if 2% of the population suffers from the allergies, how many people exactly will have it? i.e what is 2% of 1000 ?

Answer 7

Umm...40? I'm not sure I did this right

Answer 8

yes that is right

Answer 9

oops sorry that is wrong

Answer 10

dang it

Answer 11

2% of 1,000 is \(.02\times 1000=20\)

Answer 12

that is why i picked a nice round number like 1000 so the it would be easy to compute the percents

Answer 13

That what I did but my calculator said i was 40...oh well what do I have to do next?

Answer 14

ok so 20 people have the allergy, how many do not?

Answer 15

By the way that already helps a lot

Answer 16

20 out of 1000 which is... 20/10000=0.2%?

Answer 17

hold on i think i have confused you

Answer 18

we have a population of 1,000
2% have the allergy, so 20 have the allergy
if out of 1,000 people, 20 have the allergy, how many (not what percent) do not have the allergy

Answer 19

ooooh, sorry with the 10000 I got confused with another problem on my paper sorry.
Umm so 980 don't have the allergy?

Answer 20

right

Answer 21

so we know how many have it, and how many do not
now lets see how many people will test positive for it

Answer 22

of the 20 people that have it, the test is 98% accurate
what is 98% of 20?

Answer 23

btw i hope it is clear that you do \(.98\times 20\)

Answer 24

19.6?

Answer 25

right

Answer 26

yeah I'm good with the percentage....most of the time

Answer 27

i guess we have to work with the decimal, should have chosen a population of 10,000 then it would be whole numbers but too late now

Answer 28

now we know \(980\) so NOT have the allergy, lets see how many of those will also test positive

Answer 29

for them the test is 96% accurate so 96% of them will test negative, but that means 6% of them will test positive
what is 6% of 980?

Answer 30

oops bad math

Answer 31

4% will test positive
what is 4% of 980

Answer 32

39.2?

Answer 33

yeah i get that too

Answer 34

ok so now how many people total (it is a decimal) will test positive?

Answer 35

0.04 people? I'm confused

Answer 36

we computed two numbers of people that test positive right? the ones with the allergy, 19.6 and the ones without the allergy, 39.2
what is the total?

Answer 37

oh, 58.8?

Answer 38

right, that is the total number that test positive

Answer 39

out of those, we know that 19.6 actually have the allergy

Answer 40

okay

Answer 41

so the question is, IF you test positive, what is the probabilty you actually have the allergy
that is the number of people who test positive and have the allergy, divided by the total number of people who test postive
ie\[\frac{ 19.6}{58.8}\]

Answer 42

or whatever you get when you write that as a decimal, i get \(0.337\) rounded, a little more than a third

Answer 43

okay I get 0.33333etc.

Answer 44

oh maybe i put the wrong numbers in
yeah you are right

Answer 45

one third in other words

Answer 46

would you like to redo this using "baye's formula" ? will will do pretty much the same arithmetic, and get the same answer

Answer 47

yeah, it would help a lot, my teacher just throws a formula and exercises at us

Answer 48

ok do you have the formula you are supposed to use?
i can make a guess if you like

Answer 49

yeah I have it

Answer 50

go ahead and write it, we can walk through it slowly

Answer 51

It's \[P(A|B)=\frac{P(A|B)*P(A)}{ P(B)}\]

Answer 52

ok good

Answer 53

I have it right in front of me

Answer 54

first off, it this is going to make sense, you need to know what \(P(A|B)\) means
do you know what it means?

Answer 55

oh crap your definition is wrong, but we will get to that in a second

Answer 56

It means probability of A given B

Answer 57

ok good

Answer 58

and the definition (not baye's theorem) is
\[P(A|B)=\frac{P(A\cap B)}{P(B)}\]

Answer 59

My teacher made us study about a formula like that, but I really didn't get a thing

Answer 60

baye's theorem, at least one form, is not quite what you wrote, the condition is switched on the right
it is \[P(A|B)=\frac{P(B|A)*P(A)}{ P(B)}\]

Answer 61

ok lets see if we can get it

Answer 62

oh yeah my mistake

Answer 63

ok now the numerator in your fraction \[P(B|A)P(A)\] is just another way to find \(P(A\cap B)\) the probability that both A and B occur

Answer 64

That's where I get extremely confused

Answer 65

lets use our example
you want the probability you have the allergy GIVEN you test positive

Answer 66

okay

Answer 67

we can put A = you have the allergy, B = you test positive
then we are trying to find \(P(A|B)\) the probability you have the allergy given that you test positive

Answer 68

let me know if i lost you, we need to go slow i am sure

Answer 69

okay so far I think I'm okay...I think so for now

Answer 70

ok to compute that we need the numerator \(P(A\cap B) \) i.e. the probability you have it AND test positive
we were not told that number

Answer 71

so far so good

Answer 72

but we were told \(P(B)\) the probabily you have the disease, that was \(2\%=0.02\)

Answer 73

okay

Answer 74

and we were also told \(P(B|A)\) the probability you have the disease given that you test positive
that was the \(98\%=0.98\)

Answer 75

to find the numerator, multiply those together, i.e. the probability you have the disease AND test positive is the probability you have the disease times the probability you test positive GIVEN that you have the disease, i.e. \[.02\times.98=.0196\]

Answer 76

you notice that is the same numerator we had before, just with the decimal moved over three places, because we did not start out with a population of 1,000 we just went straight for the number

Answer 77

so far so good?

Answer 78

Wait why did you change the 98% to a decimal. to make it easier to work with?

Answer 79

when doing any work in math, you use numbers, not percents
\(98\%=0.98\) as a number
percents are nice to look at, but think about what you did when you wanted to find 98% of 20
you multiplied 20 by .98 right, not by 98

Answer 80

ready to find the denominator?

Answer 81

okay

Answer 82

that is \(P(B)\) the probability you test positive
remember how we found that before? we had to add two numbers, they were \(19.6\) and \(39.2\)
they will be the same this time, but with the decimal in a different place

Answer 83

the fist one is the \(.00196\) we already found, that is the number of people who test positive given that they have the disease

Answer 84

so you divided it by 1000 right?

Answer 85

we also need the number of people who test positive who do NOT have the disease

Answer 86

well, yes, but we just found again computing \(.02\times .98=.00196\)

Answer 87

to compute the probability that you test positive if you do not have the disease we used \(.04\times .980=.0392\)

Answer 88

okay

Answer 89

I'm sorry I really want to learn but I have to go to bed

Answer 90

now add together to get the probability you test positive, that is \(.00196+.00392=.00588\)

Answer 91

you did help a lot already though I'll make sure I have a lot of time next time I ask a question though

Answer 92

yeah it is late
you can look at this later and see that we are really using the same numbers as before

Answer 93

okay

Answer 94

later

Answer 95

later!