Combinatorics: Solution Explanation

[Dblooos]

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.

[theCodeToGMAT]

Duplicate Post: https://www.beatthegmat.com/combinatoric ... tml#706647

[GMATGuruNY]

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.

The logic behind the explanation is sound.

Of the 14 people, exactly 1 must receive Prograine.
Each person has an EQUAL chance of being selected.
Thus, the probability that DONALD is selected = 1/14.

Of the 14 people, exactly 1 must receive Ropecia.
Each person has an EQUAL chance of being selected.
Thus, the probability that DONALD is selected = 1/14.

Since either outcome is favorable, we ADD the fractions:
1/14 + 1/14 = 2/14 = 1/7.

Check here for a similar problem:
https://www.beatthegmat.com/board-of-dir ... 98739.html

[Brent@GMATPrepNow]

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?

We can also solve this using the complement.

P(Donald receives either Progaine or Ropecia) = 1 - P(Donald receives neither Progaine nor Ropecia)

P(Donald receives neither Progaine nor Ropecia)
At this point, I like to ask, " What exactly must occur in order for this event to happen?"
Well, for Donald to receive neither Progaine nor Ropecia it must be the case that he does not receive Prograine during the first selection and he does not receive Ropecia during the second selection.

In other words, . . .
P(Donald receives neither Progaine nor Ropecia) = P(Donald does not receive Prograine during the 1st selection AND he does not receive Ropecia during the 2nd selection.
= P(Donald does not receive Prograine during the 1st selection) x P(Donald does not receive Ropecia during the 2nd selection)
= (13/14)x(12/13)
= 12/14
= 6/7

So, P(Donald receives either Progaine or Ropecia) = 1 - (6/7)
= [spoiler]1/7[/spoiler]

Cheers,
Brent

[Scott@TargetTestPrep]

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 probability he is picked for Progaine is 1/14.

The probability he is picked for Ropecia is 13/14 x 1/13 = 1/14. (In order for him to be picked for Ropecia requires that he NOT be picked for Progaine the first round, with probability 13/14, multiplied by the probability that he IS picked for Ropecia on the second round, which is 1/13).

Therefore, the probability he is picked for either medicine is 1/14 + 1/14 = 2/14 = 1/7.

Answer: 1/7