It's a Puzzlement
Guessing May Not be the Answer
by John P. Robertson
One quarter of the point value of the question will be subtracted for each incorrect answer. No points will be added or subtracted for responses left blank." Excerpt from the CAS Exam 6 Instructions
This surprising puzzle is from Tom Struppeck.
A student taking Part 6 knows exactly what he knows, and knows what he doesn't know. He has answered every question for which he knows the answer, and he knows that he has scored exactly 53 points so far. He also knows that he needs 54 points to pass (we won't ask how he knows this). There are 20 one-point, multiple-choice questions left, each with five possible answers. He can't do any of these, and can't even eliminate any answers. He wants to guess randomly at these remaining 20 in the hopes of getting his final score up to 54 or more. That is, he wants to maximize the probability of scoring at least one point in total on these 20 one-point questions, by random guessing. How many questions should he guess at, and how many should he leave blank?
For "extra credit," suppose there are n one-point questions left and you want to maximize the probability of getting one point or more by guessing. Can you give a rule that tells how many you should guess at, and how many you should leave blank? Can you prove that rule? Can you make any general statements about maximizing the probability of getting r or more points (proof not required)?
The diagram at right shows the solution to this very difficult puzzle. Robert Ballmer, Bob Conger, and Walter Fransen submitted solutions.
We had many fine entries to the actuarial palindrome contest, which makes ranking the results very difficult. With help from Mark Saltveit of The Palindromist Magazine (www.realchange.org/pal/index2.htm), Walt Wright, and Pete Lindquist, three "first place" winners have been, somewhat arbitrarily, selected:
Bret Shroyer"NAIC, it's I, Tat," said I, a statistician.
Bob SpitzerPalindromic actuary's business card: "STATS WONK" I Know Stats
Paul IvanovskisAssess a risk, sir: assess ä.
Space does not permit printing all entries, but a few of the others submitted were:
Paul IvanovskisDr. Occam, axe dud exam? Accord!
Sums are Part 1 Trap, Erasmus.
"Ehports at a crisis?" "Si, sircatastrophe!"
Mike ZinitiA disgruntled exam taker confronts CAS President Bob Conger on the fact that he just failed: Yo, B., revel! CAS exam axes a clever boy!
Jay HallLook! Si! Risk sir is kool!
Bob Gardner (double)Part oneno trap?
NO! (ital.) Exam tenI net max elation!
Todd HubalData, et al up in a min. I manipulate a tad.
Jonathan EvansBenefits, trends, development for adjusted premiums and losses with ratemaking actuariesAll most useful!making useful most all actuaries ratemaking with losses and premiums, adjusted for development, trends, benefits.
Louis Doray, George M. Levine, Charlie Orlowicz, David Uhland, and Glenn Walker also submitted palindromes. All entries are available by clicking here or by sending an e-mail to me at firstname.lastname@example.org (specify Word or PDF). Read them all, and pick your own favorites!