Combinatorics: Solution Explanation

14 --> 1(Progaine)
13 --> 1(Ropecia)
12 --> 1

To find: Prob of Progaine or Ropecia

Probability of (Progaine) = 1/14

Probability of (Ropecia) = 13/14*1/13 = 1/14

Probability = 1/14 + 1/14 = 1/7

Dblooos wrote:A medical researcher must choose one of 14 patients to receive an experimental
medicine called Progaine. The researcher must then choose one of
the remaining 13 patients to receive another medicine, called Ropecia.
Finally, the researcher administers a placebo to one of the remaining 12
patients. All choices are equally random. If Donald is one of the 14 patients,
what is the probability that Donald receives either Progaine or Ropecia?

The solution says:

None of the 14 patients is "special" in any way, so each of them must have the same
chance of receiving Progaine or Ropecia. Since Progaine is only administered to one patient,
each patient (including Donald) must have probability 1/14 of receiving it. The same logic
also holds for Ropecia. Since Donald cannot receive both of the medicines, the desired
probability is the probability of receiving Progaine, plus the probability of receiving Ropecia:
1/14 + 1/14 = 1/7.

Is this correct? If yes how? If not what should be the correct answer? Please help. Thank you.

Hi Dbloos!

That answer is in fact correct.

Here's another way of looking at it: 2 out of the 14 patients will be chosen to receive P or R. Since the selection is random, each patient has a 2/14 chance of being one of the two.

We can confirm this answer with common sense (a very powerful and often under-utilized tool on the GMAT!): if Donald didn't have a 2/14 chance of getting exactly 1 of those two treatments, then at least one of the other 14 applicants would have to have a greater than 2/14 chance to get the probabilities to properly add up. Since the choices are completely random, that makes no sense!

A big takeaway from this question - and something that stumps a lot of people on probability questions - is that when you make selections without replacement, it's irrelevant whether you make them simultaneously or sequentially. So, if the question had said that all 3 patients were chosen at the same time, the answer would remain the same (2/14 for getting either P or R; 3/14 for getting either P, R or the placebo).

Thank you Rahul and Stuart. It makes sense now