An exam consists of 8 true/false questions. Brian forgets to

Hi All,

We're told that an exam consists of 8 true/false questions and that Brian forgets to study (so he must guess blindly on each question) and that any score above 70% is a passing grade. We're asked for the probability that Brian passes?

To start, since there are 8 questions that each have 2 possible outcomes, there are 2^8 = 256 possible arrangements of answers to those 8 questions. Since....

5/8 = 62.5%
6/8 = 75%

...Brian must get AT LEAST 6 of the 8 question correct to pass. Thus, we have to determine the total number of ways to get 6, 7 or 8 answers correct. This can be done with the Combination Formula.

8c6 = (8)(7)/(2)(1) = 28 possible ways to get 6 correct
8c7 = (8)/(1) = 8 possible ways to get 7 correct
8c8 = 1 possible ways to get 8 correct

Total = 28 + 8 + 1 = 37 of the 256 possible outcomes.

Final Answer: B

GMAT assassins aren't born, they're made,
Rich

BTGmoderatorDC wrote:An exam consists of 8 true/false questions. Brian forgets to study, so he must guess blindly on each question. If any score above 70% is a passing grade, what is the probability that Brian passes?

A) 1/16
B) 37/256
C) 1/2
D) 219/256
E) 15/16

OA B

Source: Veritas Prep

Given that the passing mark is 70%, Brian must get 70% of 8 = 5.6 questions correct. Since the number of questions is an integer, Brian must have at least 6 questions correct. Or, he must get 6, 7, or 8 questions correct.

Number of ways to get 6 out of 8 questions correct = 8C6 = 8!/(6!*2!) = 28;
Number of ways to get 7 out of 8 questions correct = 8C7 = 8!/(7!*1!) = 8;
Number of ways to get 8 out of 8 questions correct = 8C8 = 1

Total number of ways to get at least 6 questions correct = 28 + 8 + 1 = 37
Total number of ways to get any number of questions correct (None to all 8) = 2^8 = 256 (Each question has two choices: True/False)

Thus, the probability that Brian passes = 37/256

The correct answer: B

Hope this helps!

-Jay
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BTGmoderatorDC wrote:An exam consists of 8 true/false questions. Brian forgets to study, so he must guess blindly on each question. If any score above 70% is a passing grade, what is the probability that Brian passes?

A) 1/16
B) 37/256
C) 1/2
D) 219/256
E) 15/16

OA B

Source: Veritas Prep

Let's say that you have NO IDEA how to solve this question.
The great thing about probability questions is that, even if you don't know how to answer them, you can often eliminate some answers by using your gut instincts alone.

As Rich and Jay noted (above), Brian needs to correctly guess at least 6 of the 8 questions.
How likely does that FEEL?
Well, if should feel less than 50% likely, which means we can eliminate C, D and E.
Great, in 10 seconds, we're down to a 50-50 guess between A and B.

Guess A or B and move on knowing that you just added a lot of time to your "time bank" that you can devote to other questions.

BTGmoderatorDC wrote:An exam consists of 8 true/false questions. Brian forgets to study, so he must guess blindly on each question. If any score above 70% is a passing grade, what is the probability that Brian passes?

A) 1/16
B) 37/256
C) 1/2
D) 219/256
E) 15/16

OA B

Source: Veritas Prep

Since 70% of 8 is 5.6, Brian must answer at least 6 questions correctly in order to pass the exam.

The probability that he answers exactly 6 questions correctly is:

8C6 x (1/2)^6 x (1/2)^2 = 28 x (1/2)^8 = 28/256

The probability that he answers exactly 7 questions correctly is:

8C7 x (1/2)^7 x (1/2)^1 = 8 x (1/2)^8 = 8/256

The probability that he answers all 8 questions correctly is:

8C8 x (1/2)^8 x (1/2)^0 = 1 x (1/2)^8 = 1/256

Therefore, the probability that he answers at least 6 questions correctly is:

28/256 + 8/256 + 1/256 = 37/256

Answer: B