A certain disease has an incidence rate of 0.6%. If the false negative rate is 4% and the false positive rate is 4%, compute the probability that a person who tests positive actually has the disease
A certain disease has an incidence rate of 0.4%. If the false negative rate is 6% and the false positive rate is 5%, compute the probability that a person who tests positive actually has the disease.
A certain disease has an incidence rate of 0.6%. If the false negative rate is 6% and the false positive rate is 4%, compute the probability that a person who tests positive actually has the disease.
A certain disease has an incidence rate of 0.3%. If the false negative rate is 5% and the false positive rate is 4%, compute the probability that a person who tests positive actually has the disease.
To calculate the probability that a person who tests positive actually has the disease, you will need to use Bayes' theorem. Bayes' theorem allows us to update the probability of an event (in this case, having the disease) based on new information (the test result).
Let's calculate step by step:
1. Define the probabilities:
- Let P(D) be the probability of having the disease, which is equal to the incidence rate of 0.6%. So P(D) = 0.006 (0.6% expressed as a decimal).
- Let P(¬D) be the probability of not having the disease, which is equal to 1 - P(D). In this case, P(¬D) = 1 - 0.006 = 0.994.
- Let P(+) be the probability of testing positive.
- Let P(-) be the probability of testing negative.
2. Evaluate the false positive and false negative rates:
- The false negative rate of 4% means that there is a 4% chance of testing negative when a person has the disease. So P(-|D) = 0.04 (4% expressed as a decimal).
- The false positive rate of 4% means that there is a 4% chance of testing positive when a person does not have the disease. So P(+|¬D) = 0.04.
3. Calculate the probability of testing positive and having the disease:
According to Bayes' theorem, the probability of having the disease given a positive test result is given by the following formula:
P(D|+) = (P(D) * P(+|D)) / (P(D) * P(+|D) + P(¬D) * P(+|¬D))
Substituting the values we have:
P(D|+) = (0.006 * (1 - 0.04)) / (0.006 * (1 - 0.04) + 0.994 * 0.04)
= (0.006 * 0.96) / (0.006 * 0.96 + 0.040 * 0.994)
= 0.00576 / (0.00576 + 0.03976)
= 0.00576 / 0.04552
≈ 0.1266
Therefore, the probability that a person who tests positive actually has the disease is approximately 0.1266 or 12.66%.
true positive happens 96% of the time.
Multiply .006 times .96
false positive .006 times .04
add the two together.