57.6k views
4 votes
Unknown to a medical researcher, 7 out of 20 patients have a heart problem that will result in deach if they receive the test drug 5 patieats are randomiy selected to receive the drug and the rest receve a placebo. What is the probatility that less than 4 patients will die? Exaress your answer as a fraction of a decimal number rounded to four decimal places

User Sherin
by
7.0k points

1 Answer

0 votes

To calculate the probability that less than 4 patients will die, we need to consider the possible outcomes for the number of patients with a heart problem who will die when receiving the drug. We will use the binomial probability formula to calculate this.

Let's define:

n = Total number of patients selected (5 patients in this case)

p = Probability of a patient having a heart problem (7 out of 20, so p = 7/20)

q = Probability of a patient not having a heart problem (1 - p, so q = 1 - 7/20 = 13/20)

Now, we need to find the probability of 0, 1, 2, or 3 patients dying out of the 5 selected.

Probability of 0 patients dying (x = 0):

P(X = 0) = (n choose x) * p^x * q^(n-x) = (5 choose 0) * (7/20)^0 * (13/20)^(5-0)

Probability of 1 patient dying (x = 1):

P(X = 1) = (n choose x) * p^x * q^(n-x) = (5 choose 1) * (7/20)^1 * (13/20)^(5-1)

Probability of 2 patients dying (x = 2):

P(X = 2) = (n choose x) * p^x * q^(n-x) = (5 choose 2) * (7/20)^2 * (13/20)^(5-2)

Probability of 3 patients dying (x = 3):

P(X = 3) = (n choose x) * p^x * q^(n-x) = (5 choose 3) * (7/20)^3 * (13/20)^(5-3)

Now, we can calculate the probability of less than 4 patients dying (0, 1, 2, or 3):

P(X < 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)

Finally, let's calculate the value:

P(X < 4) = [(5 choose 0) * (7/20)^0 * (13/20)^5] + [(5 choose 1) * (7/20)^1 * (13/20)^4] + [(5 choose 2) * (7/20)^2 * (13/20)^3] + [(5 choose 3) * (7/20)^3 * (13/20)^2]

P(X < 4) ≈ 0.7174

Therefore, the probability that less than 4 patients will die is approximately 0.7174, rounded to four decimal places.

User CallumH
by
8.0k points