Question 1191638: MAT 145: Topics In Contemporary Math
More Probability
3) A disease has an incidence rate of 1.3%. A test for the disease has a false positive rate of 7% and a false negative rate of 2%. Find each of the following probabilities:
a) A person who tests positive has the disease.
b) A person who tests negative does not have the disease
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!
Terms to know:
- A positive result means the test claims the person has the disease.
- A negative result means the test claims the person doesn't have the disease.
- false positive = when the test says "positive" but the person does NOT have the disease in reality
- false negative = when the test says "negative" but the person does have the disease in reality
The terms "true positive" and "true negative" are the flipped counterparts of the items mentioned above.
Define these events
D = person actually has the disease
~D = person does not actually have the disease
P(D) = probability of someone actually having the disease
P(D) = 0.013
P(~D) = 1 - P(D) = 1 - 0.013 = 0.987
Further we'll define:
T = person tests positive for the disease
~T = person tests negative for the disease
P(T given ~D) = 0.07 = false positive rate
P(~T given ~D) = 1 - 0.07 = 0.93 = true negative rate
P(~T given D) = 0.02 = false negative rate
P(T given D) = 1 - 0.02 = 0.98 = true positive rate
--------------------------------------------
Part (a)
We're asked to find P(D given T)
In other words, if we know the person tested positive, we're wanting to find the value of P(D) based on this prior knowledge.
We'll use Bayes Theorem
P(D given T) = P(T given D)*P(D)/P(T)
First we need to use the law of total probability to say the following:
P(T) = P(T and D) + P(T and ~D)
P(T) = P(T given D)*P(D) + P(T given ~D)*P(~D)
P(T) = 0.98*0.013 + 0.07*0.987
P(T) = 0.08183
This leads to
P(D given T) = P(T given D)*P(D)/P(T)
P(D given T) = 0.98*0.013/0.08183
P(D given T) = 0.1556886227545
P(D given T) = 0.1557
Why is this answer fairly small? It's because the disease rate is really low (1.3%), so testing positive doesn't always lead to the person having the disease. This is why testing for rare diseases is often followed up with another test. Also, the false positive rate (7%) is fairly high.
Answer: Roughly 0.1557
--------------------------------------------
Part (b)
Now we're given that the person tested negative.
We need to find P(~D) based on the condition of ~T happening.
I.e. we need to find P(~D given ~T)
We'll use Bayes Theorem again.
But first we need to compute P(~T) through the law of total probability.
P(~T) = P(~T and D) + P(~T and ~D)
P(~T) = P(~T given D)*P(D) + P(~T given ~D)*P(~D)
P(~T) = 0.02*0.013 + 0.93*0.987
P(~T) = 0.91817
Or you could take the shortcut
P(~T) = 1 - P(T)
P(~T) = 1 - 0.08183
P(~T) = 0.91817
Then we can now use Bayes Theorem
P(~D given ~T) = P(~T given ~D)*P(~D)/P(~T)
P(~D given ~T) = 0.93*0.987/0.91817
P(~D given ~T) = 0.99971682803839
P(~D given ~T) = 0.9997
We get a really large probability because of the very low disease rate. It's more likely the person doesn't have the disease and a negative test leads to a strong probability like this.
Answer: Roughly 0.9997
|
|
|
