I seldom struggle to find a good notation for my probabilities any tip on that would be appreciated. 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?
Also I've got as answer $(0.6)(1/3)(2/3)$ I'm not really sure about the answer so any suggestion would be appreciated.
Drawing a tree diagram is useful in these cases. If you do that, the chances of error are less.
You missed out a probability of half (of getting drug A) that comes from the initial randomisation of treatment for patient 1.
So the answer should be $(\frac 12)(0.6)(\frac 13)(\frac 23) = \frac 1{15}$