Math Contest Reminder and a Probability Paradox??

There's still time to register for MathNotation's First Math contest for Grades 7-12 to be held on Tue Feb 3rd. I've decided to extend the registration to Thu Jan 29th. We've had interest expressed from high schools, middle schools, homeschooling teams, even a chapter of an honorary math fraternity! I'd like to see 2-3 more teams compete but I understand that many students and teachers are overextended at this time of year and this was on short notice. Look here for how to register.
----------------------------------------------------------------------------------

So what's the paradox in the title? To someone with a firm grasp of probability there won't be one, but the following series of questions may lead to a surprise for some students.

Overview of Problem
We have two scenarios in this investigation:
A set of five 4-choice multiple-choice questions and a set of five 5-choice multiple-choice questions. Of course the latter is typical of most standardized tests like SATs so this discussion may have relevance to many juniors right now!

Instructional Suggestion
For the following questions, ask students to first make educated guesses before attempting any calculations. The idea is to get them to trust their intuition which often is more accurate than their mathematical procedures!

Background
We know that the probability of correctly guessing, at random, the answer to a 4-choice question is 1/4 which is greater than the chance (1/5) of correctly guessing, at random, the answer to a 5-choice question. That was easy, right? When we ask questions about more than one question the situation becomes more complicated and a deeper understanding of probability concepts is needed: Multiplication of probabilities of independent events, binomial probabilities, etc...

The Investigation

(a) Which of the following is more likely? Randomly guessing all 5 wrong on a 5-choice multiple choice quiz or randomly guessing all 5 wrong on a 4-choice multiple choice quiz?
By intuition (no calculation, respond in 10 sec or less): _________________
Explanation of Intuitive Guess (this may be worthy of class discussion):
Now compute each probability and compare result to your intuitive answer.

(b) Which is more likely? Randomly guessing at least one right out of five on a 5-choice multiple-choice quiz or on a 4-choice multiple-choice quiz?
By intuition: ______________
Explanation of intuitive guess:
By calculating:

(c) How's your intuition doing so far?
Let's try this one:
Which is more likely:
Randomly guessing exactly one right out of 5 on a 5-choice quiz or on a 4-choice quiz?
By intuition:
By calculating:

Any surprises? In case your results don't agree with mine, I will tell roughly you what I got (actual probabilities below). The probability of guessing exactly one right out of five on a 5-choice quiz is slightly more than the probability of guessing exactly one right on a 4-choice quiz! A paradox? An anomaly of the arithmetic involved? Logical? Can you explain it? Try!

(d) Back to normalcy? Compute the probabilities of getting exactly two right out of five on a 5-choice quiz and on a 4-choice quiz. Has the order of the universe been restored!


Selected Answers (not the norm for this blog):
(b) Approx 67.2% on a 5-choice quiz; 76.3% on a 4-choice
(c) Approx 40.96% on a 5-choice quiz; 39.55% on a 4-choice
(d) 20.48% on a 5-choice quiz; approx 26.37% on a 4-choice
Pls check these results for accuracy!!

What are the fundamental concepts in this investigation? What are the learning benefits of this series of questions? Please understand that my intent on this blog is to suggest instructional methods, never to impose. You may find far more effective ways to convey the essential concepts here but, from my experience, there's only sure way to perfect our craft. Keep experimenting and asking questions!!