Unknown to a medical researcher, 4 out of 22 patients have a heart problem that will result in death if they recelve the test drug. 8 patients are randomly selected to recelve the drug and the rest recelve a placebo. What is the probability that more than 1 patient will die? Express your answer as a fraction or a decimal number rounded to four decimal places.

To solve this problem, we can model the situation using the hypergeometric distribution. The hypergeometric distribution is appropriate here because we are sampling without replacement from a finite population. ### Parameters: - Total number of patients (N): 22 - Number of patients with heart problems (K): 4 - Number of patients selected to receive the drug (n): 8 - Number of patients without heart problems (N - K): 22 - 4 = 18 We want to find the probability that more than 1 patient will die after receiving the drug. This means we need to calculate the probabilities for 0 and 1 patient dying and then subtract these from 1. ### Step 1: Calculate the probability of 0 patients dying To find the probability of exactly 0 patients dying, we need to select all 8 patients from the 18 who do not have heart problems. \[ P(X = 0) = \frac{\binom{K}{0} \cdot \binom{N-K}{n}}{\binom{N}{n}} = \frac{\binom{4}{0} \cdot \binom{18}{8}}{\binom{22}{8}} \] Calculating the binomial coefficients: - \(\binom{4}{0} = 1\) - \(\binom{18}{8} = \frac{18!}{8!(18-8)!} = \frac{18!}{8!10!} = 43758\) - \(\binom{22}{8} = \frac{22!}{8!(22-8)!} = \frac{22!}{8!14!} = 3191264\) Now substituting these values into the formula: \[ P(X = 0) = \frac{1 \cdot 43758}{3191264} \approx 0.0137 \] ### Step 2: Calculate the probability of 1 patient dying To find the probability of exactly 1 patient dying, we need to select 1 patient from the 4 with heart problems and 7 patients from the 18 without heart problems. \[ P(X = 1) = \frac{\binom{K}{1} \cdot \binom{N-K}{n-1}}{\binom{N}{n}} = \frac{\binom{4}{1} \cdot \binom{18}{7}}{\binom{22}{8}} \] Calculating the binomial coefficients: - \(\binom{4}{1} = 4\) - \(\binom{18}{7} = \frac{18!}{7!(18-7)!} = \frac{18!}{7!11!} = 31824\) Now substituting these values into the formula: \[ P(X = 1) = \frac{4 \cdot 31824}{3191264} \approx 0.0399 \] ### Step 3: Calculate the probability of more than 1 patient dying Now we can find the probability of more than 1 patient dying: \[ P(X > 1) = 1 - P(X = 0) - P(X = 1) \] Substituting the values we calculated: \[ P(X > 1) = 1 - 0.0137 - 0.0399 \approx 1 - 0.0536 = 0.9464 \] ### Final Answer The probability that more than 1 patient will die is approximately: \[ \boxed{0.9464} \]