A doctor is using an adaptive response technique in their clinical trial to deci

Question

A doctor is using an adaptive response technique in their clinical trial to decide whether patients get drug A or a placebo. They start with one red ball and one green ball in a bucket. For patient one, they will pull out a ball and look at the colour. If it's a red ball, the patient gets drug A. If it's a green ball, the patient gets the placebo. If the patient receives a placebo, the doctor will do nothing to the bucket before selecting for the next patient. If the patient receives drug A and has a positive outcome, they will add a red ball to the bucket. If the patient receives drug A and have a negative outcome, they will add a green ball to the bucket. The second patient will have their treatment decided based on what is in now in the bucket and we will add balls or not according to the same rule. This continues for patient 3, etc... Suppose there is a constant probability of 0.60 that any patient who receives drug A will have a positive outcome. What is the probability patient 1 gets Drug A with a positive treatment, patient 2 gets a placebo, and patient 3 gets Drug A?

Explanation / Answer

Bucket has 1 red ball and 1 green ball. Both outcomes of picking red or green ball are equally likely.

Therefore probability of picking red ball by first patient = 0.5

After picking first red ball there is only one green ball left in the bucket.

Therefore probability that first patient receives drug A treatment = 0.5

Probability that patient receiving drug A has positive outcome = 0.6

Therefore, Probability that first patient receives drug A treatment and has positive outcome = (probability that patient receives drug A treatment) X (probability that patient has positive outcome) = 0.5*0.6 = 0.3

If first patient receives drug A and has positive outcome then they will add red ball to the bucket. Green ball is there already in bucket. So now bucket has 1 red ball and 1 green ball. So probability of picking red or green ball remains same.

Second patient will get placebo if he picks green ball,

Therefore probability of picking green ball by 2nd patient = 0.5

Or probability of treating 2nd patient with placebo = 0.5

Therefore, probability that patient 1 gets drug A with a positive treatment and patient 2 gets a placebo = (Probability that patient 1 gets drug A with a positive treatment) X (probability that patient 2 gets a placebo) = 0.3*0.5 = 0.15

Since second patient gets a placebo, no ball will be added to the bucket. hence now there is a single red ball in the bucket.

So for third patient probability of selecting red ball is 1. (This is a sure event)

Therefore probability of treating third patient with drug A is 1.

Probability probability that patient 1 gets drug A with a positive treatment and patient 2 gets a placebo, patient 3 gets drug A = (probability that patient 1 gets drug A with a positive treatment and patient 2 gets a placebo) X (probability that patient 3 gets drug A) = 0.15 * 1 = 0.15

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.