Combinatorics: Solution Explanation

Question

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 · Answer

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

GMATGuruNY · Answer

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.

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

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Brent@GMATPrepNow · Answer

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.

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

Private tutor exclusively for the GMAT and GRE, with over 20 years of experience.
Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.

As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.

For more information, please email me (Mitch Hunt) at [email protected].
Student Review #1
Student Review #2
Student Review #3

Scott@TargetTestPrep · Answer

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 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

Scott Woodbury-Stewart
Founder and CEO
[email protected]

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