As Snazzlefrag indicated, your mathematician friend was mistaken. If your guesses are truly random, over the long run they will tend to be correct exactly as often as any particular answer would be. This is because the assignment of the correct answer and your guess are two independent random events. If you have Excel, you can do a little experiment to prove it to yourself.

In cell A1, type =INT(RAND()*5) -- this will represent the random correct answers on the exam. Type the same thing in cell B1 -- to represent random guesses at the right answer. In cell C1, type =IF($A1=0,1,0) -- this will generate a 1 in the cell if the correct answer was "A". Copy that to D1-G1, and modify the formula accordingly -- $A1=1, $A1=2, $A1=3, $A1=4. In cell H1, type =IF($A1=$B1,1,0) -- that will be 1 if the random guess in column B was correct. Now copy this row of cells down for 100 rows. Finally, in cell C101, type =SUM(C1:C100) and then copy that to cells D101-H101.

Now if you look at the totals in row 101, you should expect each one to be around 20 -- some will be higher, and some lower. Hit your F9 key to recalculate and see how the sums change. Notice that the random guess total varies just like any of the fixed answer totals. Sometimes it will be the highest, and sometimes it will be the lowest, but it tends to be around 20.

Having said that, it turns out that making truly random guesses is difficult for a human -- we tend to be too pattern-oriented. So it's easier and no less effective (in the long run) to stick with the same answer all the time.

-Gary-