Answer:
Approximately 0.0294.
Step-by-step explanation:
Assume that there's only one correct choice in each question.
- The chance of getting a question correct by random guess is 1/4.
- The chance of getting a question wrong by random guess is 3/4.
What's the probability that exactly 12 answers are correct?
- 12 out of the 28 answers need to be correct. [tex]\displaystyle \left(\frac{1}{4}\right)^{12}[/tex].
- The other 28 - 12 answers need to be incorrect. Multiply by [tex]\displaystyle \left(\frac{3}{4}\right)^{28 - 12}[/tex].
- There are more than one way of choosing 12 answers out of 28 without an order. Multiply by the combination "12-choose-28" [tex]\displaystyle \left(\begin{array}{c}12\\28\end{array}\right)[/tex].
The probability of getting exactly 12 answers correct is:
[tex]\displaystyle \left(\frac{1}{4}\right)^{12} \times \left(\frac{3}{4}\right)^{28 - 12}\times \left(\begin{array}{c}28\\12\end{array}\right)\approx 0.0182[/tex].
With the same logic, the probability of getting [tex]x[/tex] ([tex]x\in \mathbb{Z}[/tex], [tex]12\le x\le 28[/tex]) correct out of the 28 random answers will be
[tex]\displaystyle \left(\frac{1}{4}\right)^{x} \times \left(\frac{3}{4}\right)^{28 - x}\times \left(\begin{array}{c}28\\x\end{array}\right)[/tex].
The probability of getting at least 12 correct out of 28 random answers is the sum of
- the probability of getting exactly 12 correct out of 28, plus
- the probability of getting exactly 13 correct out of 28, plus
- the probability of getting exactly 14 correct out of 28, plus
- the probability of getting exactly 15 correct out of 28, plus
- the probability of getting exactly 16 correct out of 28, plus
- ... all the way to the probability of getting exactly 28 correct out of 28.
The Sigma notation might help:
[tex]\displaystyle \sum_{x = 12}^{28}{\left[\left(\frac{1}{4}\right)^{x} \times \left(\frac{3}{4}\right)^{28 - x}\times \left(\begin{array}{c}x\\28\end{array}\right)\right]}[/tex].
Evaluate this sum (preferably with a calculator)
[tex]\displaystyle \sum_{x = 12}^{28}{\left[\left(\frac{1}{4}\right)^{x} \times \left(\frac{3}{4}\right)^{28 - x}\times \left(\begin{array}{c}x\\28\end{array}\right)\right]} \approx 0.0294[/tex].
