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It's a Puzzlement


Easy Questions?

by John P. Robertson

How do you answer the following questions?

  1. How long did the Hundred Years War last?
  2. Which country makes Panama hats?
  3. From which animal do we get catgut?
  4. In which month do Russians celebrate the October Revolution?
  5. What is a camel's hair brush made of?
  6. The Canary Islands in the Atlantic are named after what animal?
  7. What was King George VI's first name?
  8. What color is a purple finch?
  9. Where are Chinese gooseberries from?
  10. Who is buried in Grant's Tomb?

Guessing May Not Be the Answer
The puzzlement involved multiple choice questions on a CAS exam, where one point is awarded for a correct answer, a quarter point is taken off for a wrong answer, zero points are given for an unanswered question, and there are five possible answers for each question. The main puzzlement was to determine how many questions to guess at out of 20 to maximize the probability of scoring at least one point.

Frank Baum's solution started by looking at the extremes. If you answer just one question, you have a 20 percent chance of scoring one point. But if you answer a huge number of questions, the odds of scoring one point become almost 50 percent. To see this consider that, as the number of questions answered gets large, it becomes increasingly unlikely that you will answer exactly 20 percent of them correctly. And chances are even that you will answer more than 20 percent or less than 20 percent correctly. Next, he notes that you do not want to try to pick up any more than one point. For example, if you guess at six problems, and you get two right, your net score is the one point you want. If you guess at five problems, you still need to get at least two of them right, which would give you a net score of at least 1.25. Common sense tells you that the chances of getting at least two out of six are better than the chances of getting at least two out of five. Also, if you guess at seven, you need at least three right, giving you a score of at least two. Again, the chances of getting two out of six are better than getting three out of seven. Note that if you get three right, you could guess at up to 11 questions, and still score the one point. And getting three right out of 11 is more likely than getting three right out of 7, 8, 9, or 10.

In general, local peaks in the probability of getting at least one point occur when you guess at 5n + 1 questions. So, for the problem at hand, guessing at 16 of the 20 questions is optimal, as 16 is the largest number of the form 5n + 1 less than or equal to 20. It turns out that there is one exception to this general rule, and that is that if there are five questions left, it's better to guess at all five than just one.

If you want to score r points, where r is an integer, the general rule is to guess at 5n + r questions, where n is as large as possible. But, there may be exceptions if the total number of questions available is small.

Gary Venter noted that if you want to score 0.25 points, and there are at least four questions available, your best strategy is to guess at exactly four questions. This gives a probability of getting at least 0.25 points of a bit more than 59 percent. Guessing more gives a lower probability. Gary also points out that the BETADISTR function in Excel can be used to compute the probability of getting at least r points from m questions. Let s be the least integer greater than or equal to (m + 4r)/5. Then BETADISTR(0.2, s, m - s + 1) is the required probability.

Bob Conger, Robert S. Ballmer, Andy Doll, Jon Evans, Robert Giambo, John Herder, Chris McKenna, Alex Kozmin, Jerry Miccolis, Dave Oakden, Frank Rau, Christopher Yaure, and Joshua Youdovin also sent in solutions, with most solving the more general problem.