Question

A health clinic is testing individuals for a certain disease daily. The probability that a patient will have a disease is 0.34. Answer the following questions by using discrete distributions.

a) Suppose the clinic will perform the test for a total of 16 patients. What is the probability that 9 of the patients are tested positive? What is the expected number of patients who are tested positive? (State the distribution).

b) What is the probability that the first patient with a positive test result will be found in the 5th test? Calculate V(X) (State the distribution).

Answer 1

a) The probability that 9 patients out of 16 test positive for the disease can be calculated using the binomial distribution. The expected number of patients who test positive can be calculated using the mean formula for the binomial distribution. b) The probability that the first patient with a positive test result will be found in the 5th test can be calculated using the geometric distribution. The variance of X for both distributions can be calculated using the corresponding variance formulas.

Step-by-step explanation:

a) The distribution of the number of patients who test positive for the disease out of 16 patients can be modeled by the binomial distribution. The probability that 9 patients test positive can be calculated using the binomial probability formula:

P(X = 9) = (16 choose 9) * (0.34)^9 * (1-0.34)^(16-9)

The expected number of patients who test positive can be calculated using the mean formula for the binomial distribution:

E(X) = n * p = 16 * 0.34

b) The probability that the first patient with a positive test result will be found in the 5th test can be calculated using the geometric distribution formula:

P(X = 5) = (1-0.34)^(5-1) * 0.34

The variance of X, V(X), for both distributions can be calculated using the corresponding variance formulas:

V(X) = n * p * (1- p) for the binomial distribution, and

V(X) = (1- p) / p^2 for the geometric distribution.