An Exam Consists of 8 True/False Questions GMAT Problem Solving

Question: 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?

1. \(\frac{1}{16}\)
2. \(\frac{37}{256}\)
3. \(\frac{1}{2}\)
4. \(\frac{219}{256}\)
5. \(\frac{15}{16}\)

“An Exam Consists of 8 True/False Questions” - is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

Solution and Explanation:

Approach Solution 1:
If you have a True or False question, then each question has a \(\frac{1}{2} \)chance of getting correct.
If a passing score is 70% it means Brian needs to get right or 6 questions (\(\frac{6}{8}=75 \)%), or 7 questions (\(\frac{7}{8}=87.5 \)%), or 8 questions (\(\frac{8}{8}=100 \)%).
The probability to have all 8 questions right is: \(\frac{1}{2}^8 \)
The probability to have 7 questions right and 1 wrong is:
\(C1_8*\frac{1}{2}^7*\frac{1}{2}=\frac{8}{2}^8\)(choose this incorrect answer \(C1_8\), the probability to have 7 right is \(\frac{1}{2}^7\)

and the probability to have 1 wrong is ½)
The probability to have 6 questions right and 2 wrong is:
\(C2_8*\frac{1}{2}^6*\frac{1}{2}^2=\frac{28}{2}^8\) (choose the size 2 incorrect answers \(C2_8\), the probability to have 6 right is \(\frac{1}{2}^6\), and probability to have 2 wrong is \(\frac{1}{2}^2\)

The probability is \(\frac{1}{2}^8+\frac{8}{2}^8=\frac{28}{2}^8\)
= 37/256

Answer: B

Approach Solution 2:

Since 70% is the passing rate,0.70 ( 8 questions) = 5.6 correct answers (or 6 rounded up)
We must then look for the probability of passing:
All correct answers + 7 correct answers + 6 correct answers = probability of passing.
All correct answers = \(\frac{1}{2}^8=\frac{1}{256}\)
7 correct answers = \(\frac{8C1}{\frac{1}{2}}^8= \frac{8}{256}\)
6 correct answers = \(\frac{8C2}{\frac{1}{2}}^8= \frac{28}{256}\)Hence,
\(\frac{1}{256}+\frac{8}{256}+\frac{28}{256}\)
= 37/256
Answer: B

Approach Solution 3:

It is mentioned in the question, that there are 8 questions that each has 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 a minimum of 6 out of 8 questions must be correct to pass. So, we need to determine the total number of ways to get 6, 7, or 8 answers correct.

So, the Combination Formula states:
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.

Answer: B

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