You have a 100 question multiple-choice test (multiple choice options A-D). Your statistics teacher says that every answer is the same (ie all the answers are A). Is there any way to get an expected value25% on the test?

Question

You have a 100 question multiple-choice test (multiple choice options A-D). Your statistics teacher says that every answer is the same (ie all the answers are A). Is there any way to get an expected value>25% on the test?

btaylor · Answer

If you guess that they're all A, then there is a 25% chance you will get a 100%.
If you split up your answers, then you are guaranteed a 25%. Is there any combination with more than 25%?

anonymous · Answer

acully no , the only succes propertise u have is 25% by luck :P
anless u are sure from one qs of the 100 !

anonymous · Answer

so it wud be like this
$\Huge\frac{1}{2}+\frac{1}{2}+...+\frac{1}{2}\text{ for 100 terms}$

$\Huge \text{expected} = \frac{1}{2} ×100=50 % !!!$

anonymous · Answer

unexpected to me to get that answer !

anonymous · Answer

Probably not ^.^
No matter how you split it...
say, we let
$$\Large x_n$$ be the number of questions you answered with choice (n) (here being a, b, c, and d)
etc

Then the expected score would be $$\Large \mathbb{E}=\sum_{n\in \{a,b,c,d\}} x_n\left<p(x_n)\right>$$

where $\large p(x_n)$ is the probability that the correct answer is choice n.

Anyway, that probability is presumably 25% for all four choices, so...
$$\Large \mathbb E = \frac14x_a+\frac14x_b+\frac14x_c+\frac14x_d $$

$$\Large \mathbb E = \frac14(x_a+x_b+x_c+x_d)$$
But surely, the sum of all the $\large x_n$ should be 100, since they should comprise all the questions... that's unless you intend to leave any questions blank...
$$\Large \mathbb E = \frac14(100)=25\%$$

btaylor · Answer

thank you for that thorough reply!

anonymous · Answer

^.^ 
I do try

anonymous · Answer

@PeterPan  $\color{orange}{\text AWESOME  ! !} $