Question

Upon arrival at a hospital's emergency room, patients are categorized according to their condition as critical, serious, or stable. In the past year: (i) 10% of the emergency room patients were critical; (ii) 30% of the emergency room patients were serious; (iii) the rest of the emergency room patients were stable; (iv) 40% of the critical patients died; (v) 10% of the serious patients died; and (vi) 1% of the stable patients died. Given that a patient survived, what is the probability that the patient was categorized as serious upon arrival?

(A) 0.06
(B) 0.29
(C) 0.30
(D) 0.39
(E) 0.64

Answer 1

The probability that the patient was categorized as serious upon arrival given that they survived is 0.39

Step-by-step explanation:

To find the probability that the patient was categorized as serious upon arrival given that they survived, we need to use conditional probability. Let's denote the event that a patient is serious as S and the event that a patient survived as Sc (complement of S). We want to find P(S|Sc). According to Bayes' theorem, we have:

P(S|Sc) = P(S) * P(Sc|S) / P(Sc)

We know that P(S) = 0.30, P(Sc) = 1 - P(S) = 0.70, P(Sc|S) = 1 - P(S dies|S) = 1 - 0.10 = 0.90. Substituting these values into the formula, we get:

P(S|Sc) = (0.30 * 0.90) / 0.70 = 0.39