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[Solved] A surgical unit at a major hospital performs high-risk surgery on four cardiac patients each day. Similar patients presenting with similar disease

Question: A surgical unit at a major hospital performs high-risk surgery on four cardiac patients each day. Similar patients presenting with similar disease pathology are scheduled for surgery on the same day. However on different days the patient mix can differ. As a result, the risk of mortality within 8 hours of surgery, \(X\), varies from day to day according to the probability density function

\[f(x)=\left\{\begin{array}{cc} 20 x & 0 \leq x \leq 0.1 \\ \frac{20}{9}(1-x) & 0.1 In an audit of surgical performance, a day is chosen at random by auditors to review surgical outcomes.

  1. Obtain a formula for the conditional probability of exactly three deaths within 8 hours of surgery among the four patients whose surgeries took place that day. (Assume that the number of deaths follows a Binomial distribution.)
  2. Calculate the unconditional (marginal) probability that exactly three deaths within 8 hours of surgery are observed among the four patients whose surgeries took place that day.

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