Question

Two antibiotics A and B are available as treatment for a common ear infection in children. Antibiotic A is known to cure the infection 75 % of the time. Treatment with Antibiotic A costs $ 70. Antibiotic B is known to cure the infection 90 % of the time. Treatment with Antibiotic B costs $ 85. The antibiotics work independently of one another. A health insurance company intends to recommend one of the following two plans of treatment for children with this ear infection.

Plan 1 : Treat with antibiotic A first. If it’s not effective, then treat with antibiotic B. Plan 2: Treat with antibiotic B first. If it’s not effective, then treat with antibiotic A.

a) If a doctor treats a child using plan 1, what’s the probability that the child will be cured ? b) If a doctor treats a child using plan 2, what’s the probability that the child will be cured? c) Compute the expected cost per child when plan 1 is used for treatment.
d) Compute the expected cost per child when plan 2 is used for treatment.

e) Based on the results above, which plan would you recommend? Justify your answer.

Answer 1

Given:

Two antibiotics A and B are available as treatment for a common ear infection in children.

Antibiotic A is known to cure the infection 75 % of the time. Treatment with Antibiotic A costs $ 70.

Antibiotic B is known to cure the infection 90 % of the time.

Treatment with Antibiotic B costs $ 85. The antibiotics work independently of one another.

The probability of A to successfully cure the infection is 0.75

P(A)=0.75

The probability of A not successfully cure the infection is

So (A) = 1-0.75 = 0.25

The probability of B to successfully cure the infection is 0.90

P(B)=0.90

The probability of B not successfully cure the infection is

So  (B) = 1-0.90 = 0.10

Now,

a) If a doctor treats a child using plan 1, the probability that the child will be cured is

Therefore the expected cost = 137.50

e) The probability of curing is same in both the cases.

Based on the results Plan 2 is better because plan 2 has lowers the cost of treating the child.

So we would recommend to plan 2.