Question

Heart failure is due to either natural occurrences (85%) or outside factors (15%). Outside factors are related to induced substances or foreign objects. Natural occurrences are caused by arterial blockage, disease, and infection. Suppose that 18 patients will visit an emergency room with heart failure. Assume that causes of heart failure between individuals are independent.

(a) What is the probability that three individuals have conditions caused by outside factors?
(b) What is the probability that three or more individuals have conditions caused by outside factors?
(c) What is the mean and standard deviation of the number of individuals with conditions caused by outside factors?

Answer 1

To solve this problem, we can use the binomial probability formula. The probability that three individuals have conditions caused by outside factors can be calculated using combinations. The mean and standard deviation of the number of individuals with conditions caused by outside factors can also be determined.

Step-by-step explanation:

To solve this problem, we can use the binomial probability formula. Let's define the probability of an individual having a condition caused by outside factors as p = 0.15. The total number of patients is n = 18.

(a) To find the probability that three individuals have conditions caused by outside factors, we can use the formula:

P(X = 3) = C(18, 3) * (0.15)^3 * (0.85)^15, where C(18, 3) is the number of combinations of choosing 3 patients out of 18.

(b) To find the probability that three or more individuals have conditions caused by outside factors, we can use the formula:

P(X ≥ 3) = P(X = 3) + P(X = 4) + ... + P(X = 18)

(c) The mean (expected value) of the number of individuals with conditions caused by outside factors is given by:

μ = n * p = 18 * 0.15

The standard deviation (σ) of the number of individuals with conditions caused by outside factors is given by:

σ = sqrt(n * p * (1 - p)) = sqrt(18 * 0.15 * 0.85)