Question

unknown to a medical researcher, 4 out of 21 patients have a heart problem that will result in death if they receive the test drug. 7 patients are randomly selected to receive the drug and the rest receive a placebo. what is the probability that exactly 2 patients will die? express your answer as a fraction or a decimal number rounded to four decimal places.

Answer 1

The probability that exactly 2 patients will die is approximately 0.8939.

Step-by-step explanation:

Let's solve this problem step by step.

Given:

• Total patients = 21
• Patients with heart problem = 4
• Patients without heart problem = 21 - 4 = 17
• Patients who receive drug = 7
• Patients who receive placebo = 21 - 7 = 14

We need to find the probability that exactly 2 patients will die.

To find the probability, we can use the binomial probability formula:

P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)

Where:

• P(X = k) is the probability of getting exactly k successes (in this case, k patients dying)
• n is the total number of trials (in this case, 7)
• k is the number of successful trials (in this case, 2)
• C(n, k) is the number of combinations of n items taken k at a time
• p is the probability of a single success (in this case, the probability of a patient with a heart problem dying)
• 1 - p is the probability of a single failure (in this case, the probability of a patient with a heart problem not dying)

Let's plug in the values:

P(X = 2) = C(7, 2) * (4/21)^2 * (1 - 4/21)^(7 - 2)

Calculating this expression gives us:

P(X = 2) = 21 * (4/21)^2 * (17/21)^5

= 15015/16807

≈ 0.8939

Therefore, the probability that exactly 2 patients will die is approximately 0.8939.