## UKMT Challenges: Is it beneficial to guess the answers to the last 10 questions?

In the JMC and IMC (Junior Maths Challenge and Intermediate Maths Challenge), you get 5 marks for each correct answer for questions 1-15. Since questions 16-20 are more difficult, you get 6 marks for each correct answer but -1 mark for each incorrect answer. For questions 21-25, you also get 6 marks for a right answer but -2 marks for a wrong answer. The tests are multiple choice with 5 different possible answers for each question (see question above). In the space of an hour most students manage to finish up to question 15 and then either don’t have time to answer the rest of the questions or simply don’t know how to figure out the correct answers.

So the question that I often get asked by the students is whether they should guess the answers to the last 10 questions???

Using a discrete probability distribution approach:

Finding the expected frequency of marks for questions 16-20, if you guess all five questions you can expect on average to gain an extra 2 marks. Guessing questions 21-25 would result in an average loss of 2 marks. Hence if you guessed all questions 16-25, you could expect to gain 0 extra marks. However, if you just guess 16-20 you could expect an increase of 2 marks.

Using a Binomial Distribution approach:

For questions 16-20, if you get 1 right and get the other 4 wrong then you would get 6 marks + (4 x -1 marks) = 2 marks. So if you get at least one question right, it’s worth guessing. The probability of getting at least 1 right is 67.2%.

For questions 21-25, if you get 2 right and get the other 3 wrong then you would get 12 marks + (3 x -2 marks) = 6 marks. So if you get at least 2 questions right, its again worth guessing. The probability of getting at least 2 of these questions right is 26.3%.

It seems then that guessing questions 16-20 is more likely to increase your point score then decrease it. Of course, there’ll be some students who will lose 5 marks because they weren’t lucky enough to get at least one of them right but then there’s also a small chance (0.2 x 0.2 x 0.2 x 0.2 x 0.2 = 0.032%) that a student could get all 5 questions right and gain an extra 30 marks!

If you can eliminate 2 of the answers for all of the questions 21-25, you’d have a 54% chance of improving your mark and if you could eliminate 3 of the answers then you’d have an 81.3% chance of gaining extra marks.

Now whether you would tell your students about this is another question. I personally wouldn’t advocate guessing questions. It would however be an interesting task (if not a little laborious) to get a GCSE group to draw a huge tree diagram to try to figure some of this stuff and a really awesome way for an IB student to apply their knowledge of discrete probability distributions and the binomial distribution. (Here’s a sheet with some of calculations on that you could provide as an answer sheet to an IB class).