Effectiveness of a Drug A certain disease has a mortality rate of 60%. A new drug is tested for its effectiveness against this disease. Ten patients are given the drug, and eight of them recover.

(a) Find the probability that eight or more of the patients would have recovered without the drug. (b) Does the drug appear to be effective? (Consider the drug effective if the probability in part (a) is 0.05 or less.)

(a)

To find:

The probability that eight or more of the patients would have recovered without the drug.

Given:

The number of patients is 10.

A certain disease has a mortality rate of 60%.

Approach:

Write the expression for successful outcome.

P(S)=p

Here, P(S) is the possible outcome for success.

Write the expression to find the probability of failure in a trail.

P(F)=1−pq=1−p……(1)

Here, q=P(F) is probability of failure and p is the probability of success in trail.

Write the expression to find the probability of getting exactly success in given number of independent trails of the experiment.

P(r successes in n trails)=C(n,r)prqn−r…... (2)

Here, n is number of trail and r number of success in the given experiment.

Write the formula to find the number of combinations.

C(n,r)=n!r!(n−r)!

Substitute n!r!(n−r)! for C(n,r) in equation (2).

P(r successes in n trails)=(n!r!(n−r)!)prqn−r……(3)

Calculation:

Consider the probability of success in trail for given experiment.

p=1−60%=1−60100=1−0.6=0.4

Substitute 0.4 for p in equation (1) to find q.

q=1−(0.4)=0.6

Substitute 10 for n, 8 for r, 0.4 for p and 0.6 for q in equation (3) to find the P(8 out of 10 recover).

P(8 out of 10 recover)=((10)!8!(10−8)!)(0.4)8(0.6)10−8=((10)!8!2!)(0

(b)

Whether the drug is effective or not.