Question 470895
Students will get 3 points for each correct answer; lose 1 point for each incorrect answer; and receive no points for unanswered questions. A student does not know the correct answer for one question. Is it to her advantage or disadvantage to guess an answer?
<pre>
This is an expectation problem:

The problem does not specify how many choices there are for the answer
to the question. So I will suppose there are n choices for the answer.
Then the probability of guessing correctly is {{{1/n}}} and the 
probability of guessing incorrectly is 1 - {{{1/n}}}.

Possibility              x (points)    P(x)     Expectation = x*P(x)
 
Answering correctly          3         1/n           3/n
Answering incorrectly       -1        1-1/n        -1(1-1/n)

So the total expectation on guessing is

{{{3/n-1(1-1/n)}}}
{{{3/n-1+1/n}}}
{{{4/n-1}}}

Since 0 points are gotten if the problem is left blank, we
want to know how many choices n must be in order for the
expection to be greater than 0.  So we set that > 0

{{{4/n-1>0}}}

Multiply through by n

{{{4-n>0}}}

{{{4>n}}}

So if there are less than 4 choices for the answer to each question,
it is to the student's advantage to guess.  If there are exactly 4
choices per question, then the expectation will be 0 either way.
However if there are more than 4 choices for each question, it is
better to leave the question unanswered.

Edwin</pre>