Question

P Heart failure is due to either natural occurrences

(87%) or outside factors (13%). Outside factors are related
to induced substances or foreign objects. Natural occurrences are caused by arterial blockage, disease, and infection.
Assume that causes of heart failure for the individuals are
independent.
a. What is the probability that the first patient with heart
failure who enters the emergency room has the condition due
to outside factors?
b. What is the probability that the third patient with heart
failure who enters the emergency room is the first one due to
outside factors?
c. What is the mean number of heart failure patients with
the condition due to natural causes who enter the emergency
room before the first patient with heart failure from outside
factors?

Answer 1

Explanation:

a. To find the probability that the first patient with heart failure has the condition due to outside factors, we simply use the given probability:

Probability (outside factors) = 13% = 0.13

So, the probability is 0.13 or 13%.

b. To find the probability that the third patient with heart failure who enters the emergency room is the first one due to outside factors, we can set up a geometric probability. The probability that the first patient is due to outside factors is 0.13 (as calculated in part a). The probability that the second patient is not due to outside factors (i.e., natural causes) is 0.87 (1 - 0.13).

Now, to find the probability that the third patient is the first one due to outside factors, we multiply the probability of the first patient being due to outside factors by the probability of the second patient being due to natural causes:

Probability (third patient is the first one due to outside factors) = 0.13 * 0.87 = 0.1131

So, the probability is approximately 0.1131 or 11.31%.

c. To find the mean number of heart failure patients with the condition due to natural causes who enter the emergency room before the first patient with heart failure from outside factors, we can use the concept of a geometric distribution.

The probability of a patient having heart failure due to natural causes is 87% or 0.87 (as given). The probability of the first patient being due to outside factors is 13% or 0.13 (as calculated in part a).

The mean (expected) number of patients with heart failure due to natural causes before the first patient with outside factors can be calculated using the formula:

Mean = 1 / Probability (outside factors)

Mean = 1 / 0.13

Mean ≈ 7.6923

So, the mean number of heart failure patients with the condition due to natural causes who enter the emergency room before the first patient with heart failure from outside factors is approximately 7.6923.