A certain disease has an incidence rate of 0.5%. If the false negative rate is 7% and the false positive rate is 2%, compute the probability that a person who tests positive actually has the disease.
Well, well, well, isn't statistics just a barrel of laughs? Let's dive in and have some fun with probabilities!
To calculate the probability of a person actually having the disease after testing positive, we need to utilize Bayes' theorem. It goes a little something like this:
P(A|B) = (P(B|A) * P(A)) / P(B)
Where:
P(A|B) - the probability of A given B
P(B|A) - the probability of B given A
P(A) - the probability of A
P(B) - the probability of B
In our case, A represents having the disease, and B represents testing positive. So let's plug in the numbers!
P(A) = 0.005 (incidence rate of the disease)
P(B|A) = 1 - 0.07 = 0.93 (1 - false negative rate)
P(B) = (0.005 * 0.93) + (0.995 * 0.02) ≈ 0.025 (probability of testing positive)
Now we can calculate the probability of having the disease given a positive test result:
P(A|B) = (0.93 * 0.005) / 0.025 ≈ 0.186 or 18.6%
So, if my calculations didn't get lost in the statistical circus, there's about an 18.6% chance that a person who tests positive actually has the disease. Don't worry, though – if you want to double-check my math, I'll be here juggling numbers and silly jokes!
To compute the probability that a person who tests positive actually has the disease, we can use Bayes' theorem.
Let's define the following:
A: Person has the disease
B: Person tests positive
We are given the following probabilities:
P(A) = 0.005 (incidence rate)
P(B|A') = 0.02 (false positive rate)
P(B'|A) = 0.07 (false negative rate)
We want to find P(A|B), the probability that a person has the disease given that they test positive.
Bayes' theorem states:
P(A|B) = (P(B|A) * P(A)) / P(B)
We can calculate the denominator P(B) using the law of total probability:
P(B) = P(B|A) * P(A) + P(B|A') * P(A')
Let's substitute the given values into the formula:
P(B) = (0.02 * 0.005) + (0.07 * (1 - 0.005))
Simplifying further:
P(B) = 0.0001 + 0.06965
P(B) = 0.06975
Now, let's calculate P(A|B):
P(A|B) = (0.02 * 0.005) / 0.06975
Simplifying further:
P(A|B) = 0.0001 / 0.06975
P(A|B) ≈ 0.001432
Therefore, the probability that a person who tests positive actually has the disease is approximately 0.001432, or 0.1432%.
To determine the probability that a person who tests positive actually has the disease, we can use Bayes' theorem. Bayes' theorem describes how to update a probability based on new evidence.
Let's break down the information given:
Incidence rate of the disease: 0.5% or 0.005
False negative rate: 7% or 0.07
False positive rate: 2% or 0.02
We need to calculate the probability of having the disease given a positive test result. Let's denote the events as follows:
A: Having the disease
B: Testing positive
We need to calculate P(A | B), the probability of having the disease given a positive test result. According to Bayes' theorem:
P(A | B) = [P(B | A) * P(A)] / P(B)
To calculate P(B | A), this represents the probability of testing positive given that the person has the disease, which is equal to 1 - false negative rate:
P(B | A) = 1 - 0.07 = 0.93
P(A) is the incidence rate of the disease:
P(A) = 0.005
P(B) can be calculated using the Law of Total Probability, considering both true and false positive cases:
P(B) = P(B | A) * P(A) + P(B | ¬A) * P(¬A)
P(B | ¬A) represents the probability of testing positive given that the person does not have the disease, which is equal to the false positive rate:
P(B | ¬A) = 0.02
P(¬A) is the complement of P(A), representing the probability of not having the disease:
P(¬A) = 1 - P(A) = 1 - 0.005 = 0.995
Now we can calculate P(B):
P(B) = P(B | A) * P(A) + P(B | ¬A) * P(¬A)
= 0.93 * 0.005 + 0.02 * 0.995
≈ 0.00465 + 0.0199
≈ 0.02455
Finally, we can calculate P(A | B) using Bayes' theorem:
P(A | B) = [P(B | A) * P(A)] / P(B)
= (0.93 * 0.005) / 0.02455
≈ 0.00465 / 0.02455
≈ 0.1891
Therefore, the probability that a person who tests positive actually has the disease is approximately 0.1891 or 18.91%.