A medical test has a 95% accuracy of detecting a Condition Z if the person has it. It also has a 97% chance to indicate that the person does not have the condition if they really don't have it. If the incidence rate of this disease is 10 out of every 100:

What is the probability that a person chosen at random will both test positive and actually have the disease (i.e., get a true positive)?

What is the probability that a person chosen at random will test positive but not have the disease (i.e., get a false positive)?

Question

A medical test has a 95% accuracy of detecting a Condition Z if the person has it. It also has a 97% chance to indicate that the person does not have the condition if they really don't have it. If the incidence rate of this disease is 10 out of every 100: 

What is the probability that a person chosen at random will both test positive and actually have the disease (i.e., get a true positive)? 

What is the probability that a person chosen at random will test positive but not have the disease (i.e., get a false positive)?

anonymous · Answer

@SolomonZelman how about dis 1?

anonymous · Answer

@DanJS halp me plz

misty1212 · Answer

if this gets confusing, pick a number of people and work with that number
make it a large power of ten

misty1212 · Answer

say 10,000 people 
how many have the disease?

anonymous · Answer

100?

misty1212 · Answer

ten out of a hundred is ten percent

anonymous · Answer

so 1000

misty1212 · Answer

ok so we know 1,000 have the disease
how many do not?

anonymous · Answer

5% of those?

danjs · Answer

sorry i'm late,, misty gots it for ya..

anonymous · Answer

it's ok

misty1212 · Answer

you are thinking too hard 
there are 10,000 total (we made that up) 
1,000 have the disease
how many people don't have it?

anonymous · Answer

9000

misty1212 · Answer

that makes more sense 
now lets compute some stuff

misty1212 · Answer

of the 1,000 that have it, how many test positive?

anonymous · Answer

95%

misty1212 · Answer

yes, 95%  of them
the question is not "what percent"  of them but "how many" of them

misty1212 · Answer

i.e. what is 95% of 1,000?

anonymous · Answer

950

misty1212 · Answer

right exactly 
so now we are almost done with the first question 
out of the 10,000 people we know that 950 of them have the disease and also test positive 
making the probability the ratio $\frac{950}{10,000}$

anonymous · Answer

9.5%?

misty1212 · Answer

that would be $.095$ or $9.5\%$ 
we could have arrived at this answer by simply computing 
$$0.1\times .95$$ as well

misty1212 · Answer

the second one is a bit harder

misty1212 · Answer

well not really 
how many people do not have the disease?  we already  found that, it is 9,000
out of those how many test positive?

anonymous · Answer

95% or 8550

misty1212 · Answer

lets back up and go slow

misty1212 · Answer

there are 9,000 people who do NOT have the disease right?

anonymous · Answer

uhh 450

misty1212 · Answer

this the question states 
 It also has a 97% chance to indicate that the person does NOT have the condition if they really don't have it.

misty1212 · Answer

if it is right 97% of the time, it must be wrong what percent of the time?

anonymous · Answer

is it 270

misty1212 · Answer

yes 3% of 9,000 is 270

misty1212 · Answer

so the answer to the second part, the probability it is a false positive, is $\frac{270}{10,000}$

anonymous · Answer

so a is 9.5% and b is 2.7%?

misty1212 · Answer

yes it is

anonymous · Answer

ok thank you, i have another problem after this, it is the last one

misty1212 · Answer

we could have got B by computing 
$$0.03\times 09$$ also

misty1212 · Answer

but it makes more sense i think if you do it with numbers