Suppose that a drug is known to be 90% effective in treating a certain disease. What is the probability that it will be successful in treating seven to ten out of 12 patients with the disease?

Probability of treating r number of patients successfully:

P(r) = (nCr)(p^r)(q^r)

Where n = Total number of patients = 12
p = probability of success = 0.90 (90%)
q = probability of failure = 0.10 (10%)

Take the different probabilities for r = 7,8,9,10 and add them up for the answer.

and

nCr = n!/[(n-r)! r! ] the binomial coefficient

To calculate the probability that the drug will be successful in treating seven to ten out of 12 patients, we can use the binomial probability formula.

The binomial probability formula is: P(x) = C(n, x) * p^x * (1-p)^(n-x)

In this case:
- n is the total number of patients (12)
- x is the number of successful treatments (from 7 to 10)
- p is the probability of success (90% or 0.9)

Now we can calculate the probability.

For x = 7:
P(7) = C(12,7) * (0.9)^7 * (1-0.9)^(12-7)

Using a calculator or software, we can calculate C(12,7) = 792.

P(7) = 792 * (0.9)^7 * (0.1)^5

We repeat this calculation for x = 8, 9, and 10.

For x = 8:
P(8) = C(12,8) * (0.9)^8 * (1-0.9)^(12-8)

Again using a calculator or software, we can calculate C(12,8) = 495.

P(8) = 495 * (0.9)^8 * (0.1)^4

For x = 9:
P(9) = C(12,9) * (0.9)^9 * (1-0.9)^(12-9)

C(12,9) = 220.

P(9) = 220 * (0.9)^9 * (0.1)^3

For x = 10:
P(10) = C(12,10) * (0.9)^10 * (1-0.9)^(12-10)

C(12,10) = 66.

P(10) = 66 * (0.9)^10 * (0.1)^2

Finally, we sum up the probabilities of getting 7, 8, 9, or 10 successful treatments:

P(7 to 10) = P(7) + P(8) + P(9) + P(10)

Calculating these probabilities will give us the answer.