Heart failure is due to either natural occurrences (87%) or outside factors (13%

Question

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 fourth patient with heart failure who enters the emergency room is the first one due to outside factors?
c) What is the probability that the fifth patient with heart failure who enters the emergency room is the third one due to outside factors?

Explanation / Answer

(a) Required probability = 0.13

(b) For the fourth patient to be first one due to outside factor, the first three should be due to natural occurrence.

Thus, required probability = .87 × .87 × .87 × .13 = .0856

(c) For the fifth patient to be third due to outside factors, the remaining two should be due to natural occurrence.

The order of the first four patients should also be considered

Thus, total probability = (0.87)^2 × (0.13^2) × 4!/(2! × 2!) × 0.13

= 0.01

(Note: 4!/(2!×2!) because from first 4 patients 2 were due to natural occurrence and two due to outside factors)

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