Question

At a hospital's emergency room, patients are classified and 20% of them are critical, 30% are serious, and 50% are stable. Of the critical ones, 30% die; of the serious, 10% die; and of the stable, 1% die. Given that a patient dies, what is the conditional probability that the patient was classified as critical?

Answer 1

The conditional probability that a patient was classified as critical given they died is approximately 66.32%.

Step-by-step explanation:

To determine the conditional probability that a patient was classified as critical given they died, we can use Bayes' theorem. First, let's calculate the overall probability of a patient dying.

The probability of a critical patient dying is 20% (probability of being critical) × 30% (probability of dying given critical) = 0.06 or 6%.

The probability of a serious patient dying is 30% × 10% = 0.03 or 3%.

The probability of a stable patient dying is 50% × 1% = 0.005 or 0.5%.

The total probability of a patient dying is the sum of these probabilities: 6% + 3% + 0.5% = 9.5%.

Now, using Bayes' theorem, we find the probability of a patient being critical given they died:

P(Critical | Died) = (P(Died | Critical) × P(Critical)) / P(Died) = (0.30 × 0.20) / 0.095 = 0.063 / 0.095 ≈ 0.6632 or 66.32%.