A new test has been developed to detect a particular type of cancer. The test must be evaluated before it is put into use. A med
ical researcher selects a random sample of 1,000 adults and finds (by other means) that 4% have this type of cancer. Each of the 1,000 adults is given the new test, and it is found that the test indicates cancer in 99% of those who have it and in 1% of those who do not.
a) Based on these results, what is the probability of a randomly chosen person having cancer given that the test indicates cancer?
b) What is the probability of a person having cancer given that the test does not indicate cancer?
a) Based on these results, what is the probability of a randomly chosen person having cancer given that the test indicates cancer?
b) What is the probability of a person having cancer given that the test does not indicate cancer?
1 answer:
The tree diagram of the problem above is attached
There are four outcomes of the two events,
First test - Cancer, Second Test - Cancer, the probability is 0.0396
First test - Cancer, Second Test - No Cancer, the probability is 0.0004
First test - No Cancer, Second Test - There is cancer, the probability is 0.0096
First test - No cancer, Second Test - No cancer, the probability is 0.9054
The probability of someone picked at random has cancer given that test result indicates cancer is
The probability of someone picked at random has cancer given that test result indicates no cancer is
There are four outcomes of the two events,
First test - Cancer, Second Test - Cancer, the probability is 0.0396
First test - Cancer, Second Test - No Cancer, the probability is 0.0004
First test - No Cancer, Second Test - There is cancer, the probability is 0.0096
First test - No cancer, Second Test - No cancer, the probability is 0.9054
The probability of someone picked at random has cancer given that test result indicates cancer is
The probability of someone picked at random has cancer given that test result indicates no cancer is
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Answer:
Step-by-step explanation:
We can use Part I of the Fundamental Theorem of Calculus:
Since we have two functions as the limits of integration, we can use one of the properties of integrals; the additivity rule.
The Additivity Rule for Integrals states that:
We can use this backward and break the integral into two parts. We can use any number for "b", but I will use 0 since it tends to make calculations simpler.
We want the variable to be the top limit of integration, so we can use the Order of Integration Rule to rewrite this.
The Order of Integration Rule states that:
We can use this rule to our advantage by flipping the limits of integration on the first integral and adding a negative sign.
Now we can take the derivative of the integrals by using the Fundamental Theorem of Calculus.
When taking the derivative of an integral, we can follow this notation:
- where u represents any function other than a variable
For the first term, replace with
, and apply the chain rule to the function. Do the same for the second term; replace
Simplify the expression by distributing and
inside their respective parentheses.
Rearrange the terms to be in order from the highest degree to the lowest degree.
This is the derivative of the given integral, and thus the solution to the problem.