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. Assume that causes of heart failure between individuals are independent.
(a) What is the probability that the 1st patient with heart failure that enters the emergency room has the condition due to outside factors?
(b) What is the probability that 3rd patient with heart failure that 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 that enter the emergency room until the 1st patient with heart failure from outside factors?
Well
To answer these questions, we need to use the concept of probability and basic counting principles.
(a) What is the probability that the 1st patient with heart failure that enters the emergency room has the condition due to outside factors?
The probability that the first patient has the condition due to outside factors is given as 15%. Since the causes of heart failure are independent between individuals, this probability remains constant for each patient. Therefore, the probability is 0.15 or 15%.
(b) What is the probability that the 3rd patient with heart failure that enters the emergency room is the first one due to outside factors?
To find the probability that the third patient is the first one due to outside factors, we need to consider the probabilities of the previous two patients having heart failure due to natural causes.
The probability that the first two patients have heart failure due to natural causes is 0.85 * 0.85 = 0.7225 or 72.25%. The remaining probability is for the third patient to have heart failure due to outside factors.
Therefore, the probability that the third patient is the first one due to outside factors is 1 - 0.7225 = 0.2775 or 27.75%.
(c) What is the mean number of heart failure patients with the condition due to natural causes that enter the emergency room until the 1st patient with heart failure from outside factors?
To find the mean number, we need to consider the probabilities of the patients having heart failure due to natural causes until the first patient with heart failure due to outside factors.
The probability that the first patient has heart failure due to natural causes is 0.85. The probability that the second patient has heart failure due to natural causes is 0.85 * 0.85, and so on.
The mean number is then the sum of the probabilities weighted by the number of patients:
1 * 0.85 + 2 * (0.85 * 0.85) + 3 * (0.85 * 0.85 * 0.85) + ...
We can calculate this using the formula for the sum of an infinite geometric series:
mean number = 1 * 0.85 + 2 * 0.85^2 + 3 * 0.85^3 + ...
Using the formula, the mean number is approximately 3.333.
Therefore, the mean number of heart failure patients with the condition due to natural causes that enter the emergency room until the first patient with heart failure from outside factors is approximately 3.333.