Question

A nurse observes that a patient in her care is experiencing symptoms of two different diseases that are considered to be independent of one another. Based on the patient's history, the probability that the patient is suffering from disease x is 0.84 and the probability that the patient is suffering from disease y is 0.76. Find the probability that the patient is suffering from: (a) both diseases (b) neither disease (c) only one disease (d) at least one disease?

1) P(both diseases) = ?
2) P(neither disease) = ?
3) P(only one disease) = ?
4) P(at least one disease) = ?

Answer 1

To find the probabilities in this scenario, we multiply the probabilities of each disease to find the probability of both diseases, subtract the sum of probabilities of each disease from 1 to find the probability of neither disease, subtract the probability of both diseases from the sum of probabilities of each disease to find the probability of only one disease, and subtract the probability of neither disease from 1 to find the probability of at least one disease.

Step-by-step explanation:

In this case, we can use the formula for calculating probabilities of independent events. Let's denote disease X as event A and disease Y as event B.

a) To find the probability that the patient is suffering from both diseases, we multiply the probabilities of each disease: P(A and B) = P(A) * P(B) = 0.84 * 0.76 = 0.6384.

b) To find the probability that the patient is suffering from neither disease, we subtract the sum of probabilities of having each disease from 1: P(neither A nor B) = 1 - (P(A) + P(B)) = 1 - (0.84 + 0.76) = 1 - 1.6 = -0.6. However, probabilities cannot be negative, so the probability of neither disease is 0.

c) To find the probability that the patient is suffering from only one disease, we subtract the probability of both diseases from the sum of the probabilities of each disease: P(only A or only B) = P(A) + P(B) - P(A and B) = 0.84 + 0.76 - 0.6384 = 0.9616.

d) To find the probability that the patient is suffering from at least one disease, we subtract the probability of neither disease from 1: P(at least one A or B) = 1 - P(neither A nor B) = 1 - 0 = 1.