Consider a medical screening excersis for a disease that in the case that indicte that 5 people of 1000 people screenedhave the disease.the screening process is known to give a false positive 3% at times and a false negative 1% at times.what is the probability that a random chosen person
A.who text positive for the disease has the disease
B.who test positive for the disease does not have thE disease
C.who test negative for the disease does not have the disease
D.who test positive for the disease indeed has the disease
To calculate the probabilities, we will use the concepts of sensitivity, specificity, prevalence, and probability calculations.
Given:
- Disease prevalence: 5 people out of 1000 have the disease, which means the prevalence is 0.005 (5/1000).
- False positive rate (also known as the complement of specificity): 3% or 0.03.
- False negative rate (also known as the complement of sensitivity): 1% or 0.01.
Let's calculate the probabilities now:
A. Probability that a randomly chosen person who tests positive for the disease actually has the disease (Positive Predictive Value):
To calculate this, we can use Bayes' Theorem.
P(Disease|Positive) = (P(Positive|Disease) * P(Disease)) / P(Positive)
P(Positive|Disease) = 1 - False negative rate = 1 - 0.01 = 0.99
P(Disease) = Prevalence = 0.005
P(Positive) = P(Positive|Disease) * P(Disease) + P(Positive|No Disease) * P(No Disease)
P(Positive) = (0.99 * 0.005) + (0.03 * (1 - 0.005))
Now, substitute the values back into Bayes' Theorem:
P(Disease|Positive) = (0.99 * 0.005) / P(Positive)
B. Probability that a randomly chosen person who tests positive for the disease does not have the disease (False Positive Rate):
To calculate this, we need to find the complement of the Positive Predictive Value, which is 1 minus the Positive Predictive Value.
P(Not Disease|Positive) = 1 - P(Disease|Positive)
C. Probability that a randomly chosen person who tests negative for the disease does not have the disease (True Negative Rate):
Since the test result is negative, this probability is simply the specificity (complement of the false positive rate):
P(Not Disease|Negative) = Specificity = 1 - False positive rate = 1 - 0.03 = 0.97
D. Probability that a randomly chosen person who tests positive for the disease indeed has the disease (True Positive Rate or Sensitivity):
This is the same as the Positive Predictive Value calculated in A.
P(Disease|Positive) = P(Positive|Disease) * P(Disease) / P(Positive)
Calculate P(Positive|Disease) and P(Disease) using the given values.
Now you can use these calculations to determine the probabilities.