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.