Question

Two antibiotics are available as treatment for a common ear infection in children. • Antibiotic A is known to effectively cure the infection 60 percent of the time. • Antibiotic B is known to effectively cure the infection 90 percent of the time. The antibiotics work independently of one another. Both antibiotics can be safely administered to children. A health insurance company intends to recommend one of the following two plans of treatment for children with this ear infection. • Plan I: Treat with antibiotic A first. If it is not effective, then treat with antibiotic B. • Plan II: Treat with antibiotic B first. If it is not effective, then treat with antibiotic A. (a) If a doctor treats a child with an ear infection using plan I, what is the probability that the child will be cured? (b) If a doctor treats a child with an ear infection using plan II, what is the probability that the child will be cured?

Answer 1

To find the probability that a child will be cured using Plan I, multiply the probabilities of each antibiotic being effective. For Plan II, multiply the probabilities in the opposite order. The probability of a child being cured using Plan I is 6%. The probability of a child being cured using Plan II is 36%.

Step-by-step explanation:

To find the probability that a child will be cured using Plan I, we need to consider the probabilities of each antibiotic being effective.

Antibiotic A cures the infection 60% of the time, so the probability of it not being effective is 40% (100% - 60%). If Antibiotic A is not effective, then we move on to Antibiotic B. Antibiotic B cures the infection 90% of the time, so the probability of it not being effective is 10% (100% - 90%).

The overall probability of a child being cured using Plan I is the product of the probabilities of each step. So the probability is 60% (for Antibiotic A being effective) multiplied by 10% (for Antibiotic B being effective after Antibiotic A is not effective), which equals 6%.

Therefore, the probability that a child will be cured using Plan I is 6%.

To find the probability that a child will be cured using Plan II, we follow a similar process. Antibiotic B cures the infection 90% of the time, so the probability of it not being effective is 10%. If Antibiotic B is not effective, then we move on to Antibiotic A. Antibiotic A cures the infection 60% of the time, so the probability of it not being effective is 40%.

The overall probability of a child being cured using Plan II is the product of the probabilities of each step. So the probability is 90% (for Antibiotic B being effective) multiplied by 40% (for Antibiotic A being effective after Antibiotic B is not effective), which equals 36%.

Therefore, the probability that a child will be cured using Plan II is 36%.

Answer 2

Step-by-step explanation:

the probability of A working is 60% or 0.6, so the probability it does not work is 40% or 0.4.

the probability of B working is 90% or 0.9, so the probability it does not work is 10% or 0.1.

plan 1

it is the probabilty that either A works, or (if it is not working) B works.

0.6 + 0.4×0.9 = 0.6 + 0.36 = 0.96

plan 2

either B works, or (if it is not working) A works

0.9 + 0.1×0.6 = 0.9 + 0.06 = 0.96

both plans have the same success probability.