Welcome to an example
of conditional probability.
We'll first determine the probability using a table
and then also find the probability
using the formula for Bayes' Theorem.
"A certain disease has an incidence rate of 0.7%.
If the false negative rate is 2.5%
and the false positive rate is 1.5%,
compute the probability that a person who tests positive
actually has the disease."
Which means we are looking for the conditional probability,
the probability a person has a disease
given they test positive.
To complete the table, we will assume 100,000 people.
So to begin, because the disease has
an incidence rate of 0.7%, we need to find 0.7% of 100,000.
To do this we convert 0.7% to a decimal
and multiply by 100,000,
which gives us 0.007 times 100,000
which is equal to 700.
So the total for the row that represents those
that have the disease is 700,
which should be this cell here.
Which means the number of people that don't have the disease
or no disease would have to be 100,000 minus 700,
which is 99,300.
This is the total for the number of people
that have no disease.
Next, the false negative rate is 2.5%.
False negative means the person test negative,
but actually has a disease.
And therefore the next step is
to find 2.5% of the 700 people that have the disease.
So 2.5% of 700
is equal to 0.025 times 700,
which is equal to 17.5.
So 17.5 represents the number of people that test negative,
but actually have disease,
which goes in this cell here.
And therefore, the number of people that have the disease
that test positive would have to be 700 minus 17.5,
which is 682.5.
Now, I know we can't have a decimal part of a person,
but to model this situation
we are going to use decimals in the table.
Again, we obtained 682.5
by taking the total number of diseased people of 700
and subtracting those that tested negative, or 17.5.
Next we're told the false positive rate is 1.5%.
False positive means they tested positive for the disease,
but did not have the disease,
which means the next step is to find 1.5%
of the people with no disease, or 1.5% of 99,300,
which is equal to 0.015 times 99,300,
which is equal to 1489.5.
So again, this would be the number of people
that test positive that don't have the disease,
which would be the amount in this cell here.
And therefore, the number of people
that don't have the disease that test negative
would have to be 99,300 minus 1489.5,
which is equal to 97,810.5.
Again, this value came from the total of people
with no disease of 99,300
and subtracting those
that have no disease that test positive, which is 1489.5.
The next step is to sum the columns,
the positive column,
the negative column, and the total column.
Well, we know the sum of the total column
is going to be 100,000.
The sum of the positive column is 2,172.
And the total of the negative column is 97,828.
If we total this last row, we'll also get a sum of 100,000.
And now we have the information from the table we need
in order to find the probability
a person has a disease given they test positive.
Because it's given they test positive,
we are only concerned about the positive column.
Of the 2,172 people that tested positive,
682.5 actually have the disease,
which means the probability a person has a disease
given they test positive is equal to 682.5 divided by 2,172,
which as a decimal rounded
to four decimal places, is 0.3142,
which as a percent to two decimal places is 31.42%.
And now let's find the same probability
using the formula for Bayes' Theorem on the next slide.
So the first thing to recognize when applying the formula
is event A in our case is the event
of having the disease,
and event B is the event of testing positive.
So applying the formula for Bayes Theorem,
the probability of A is the probability
of having the disease.
The probability of B given A is the probability
of testing positive given they have the disease.
In the denominator, and again,
we have the probability of disease times the probability
of testing positive given they have the disease.
Notice how we do have some repetition
in the numerator and denominator,
here in this product, as well as this product.
And then we have plus the probability of not A,
which is the probability of no disease
times the probability of B given not A,
which is the probability of testing positive
given no disease.
And now let's find these possibilities.
In the numerator we have the probably of disease,
which is the incidence rate of 0.7%, or as a decimal, 0.007,
and then we have times the probably of testing positive
given they have the disease.
This is going to be equal to 100%
minus the false negative rate.
Remember, the false negative rate is the percent
that test negative that actually have the disease,
which means 100% minus 2.5%, or 97.5%,
gives us the probability a person test positive
given they have the disease.
97.5% as a decimal is 0.975.
In the denominator, again,
we have the probability of disease,
which is, again, 0.007 times,
again, the probability of testing positive
given they have the disease, which again is 100% minus 2.5%,
or as a decimal, 0.975.
And then we have plus the probability of no disease,
which is equal to 100% minus the incidence rate of 0.7%,
which is 99.3%, or as a decimal, 0.993.
And then we have times the probability of testing positive,
but given no disease.
Well, this is going to be the false positive rate of 1.5%.
Remember, a false positive means they test positive,
but don't have the disease.
So we have times 0.015
and now let's go to the calculator
and verify we get the same result.
And notice, we do get the same value.
To four decimal places we have 0.3142.
Which of course is equal to 31.42%,
the same that we found using the table.
So these types of problems can be solved
using just the table or just Bayes' Theorem.
I hope you found this helpful.