Priorities and Risk – What Does a Student Do?

Suppose you are studying for two exams, one of which is much harder than the other, but which are both equally important to you. You estimate you have a 45 percent chance of passing the easy exam, but only a 5 percent chance of passing the difficult one. You have enough money to pay for a tutor to help you prepare for one of the exams, and you estimate that the tutor could boost your chances of passing by around 5 percent. which exam should you spend your money on? 

Dylan Evans poses this question in his new book Risk Intelligence (pages 154-55). He thinks the right answer to this question is that it makes no difference:

A person with high risk intelligence would feel indifferent; it wouldn’t matter to her which exam she spent her money on. For her, a 5 percent improvement is a 5 percent improvment, and that’s that. 

However, I think he’s wrong about this. The two outcomes are not identical, and the choice depends on the student’s priorities. Specifically, it depends on whether the student wants to maximize the chance of passing at least one exam, or maximise chance of passing both exams. I contacted Dylan with my own analysis and he invited me to present it here in the Projection Point blog.

In the scenario a student is studying two exams, one hard, one easy. She can get tutoring in one of the subjects and increase her chances of passing that exam by 5%. The exams are equally important. Does it matter in which exam she receives extra tutoring? The text suggests that her probability of passing the two exams changes from (0.45, 0.05) to either (0.5, 0.05) or (0.45, 0.1), i.e. by five percentage points. The analysis states that a highly risk intelligent student should be indifferent to the two options – after all five percent is five percent. I wasn’t so sure and decided to work out the example fully.

If the student gets tutoring in the easier exam:

  • Probability of passing both = (0.45 + 0.05) * 0.05 = 0.025
  • Probability of failing both = (1-(0.45+0.05))*(1-0.05) = 0.475
  • Probability of passing one or more = 1 – prob of failing both = 1-0.475 = 0.525

If the student gets tutoring in the harder exam:

  • Probability of passing both = (0.45) * (0.05 + 0.05) = 0.045
  • Probability of failing both = (1-0.45)*(1-(0.05+0.05)) = 0.495
  • Probability of passing one or more = 1 – prob of failing both = 1-0.495 = 0.505

So the outcomes are different and the choice depends on whether it is more important to increase the chances of passing at least one exam, or more important to try and pass both exams. For example both of the exams might be needed to pass the year, or alternatively only one successful exam is needed to pass the year. Note that the two exams are still equally important.

For these two different priorities the student should choose as follows:

  1. Need to pass both exams – get tutoring in the hard exam
  2. Need to pass one (or more) exams – get tutoring in the easy exam

Benefit of tutoring is proportional to current probability of passing

On the other hand if the benefit of getting tutoring is an increase in the probability of passing by a factor of 0.05, rather than by 5 percentage points, then there is no difference in outcomes. The 0.45 probability of passing goes to 0.45*1.05 or the 0.05 probability goes to 0.05*1.05.

If the student gets tutoring in the easier exam:

  1. Probability of failing both = (1-(0.45*1.05))*(1-0.05) = 0.501125
  2. Probability of passing both = (0.45 *1.05) * 0.05 = 0.023625
  3. Probability of passing one or more = 1 – prob of failing both = 1 – 0.501125 = 0.498875

If the student gets tutoring in the harder exam:

  1. Probability of failing both = (1-0.45)*(1-(0.05*1.05)) = 0.521125
  2. Probability of passing both = (0.45) * (0.05 * 1.05) = 0.023625
  3. Probability of passing one or more = 1 – prob of failing both = 1 – 0.501125 = 0.498875

Finally, when I contacted Dylan he suggested another possibility; if the student wanted to maximise the combined mark of the two exams, and if the expected sum of the marks in the exam, without tutoring, is (0.45m + 0.05m = 0.5m) then tutoring will raise that to either (0.5m + 0.05m) or (0.45m + 0.1m) which is (0.55m) in both cases.

Summary

  • The priorities of the student matter. These include:
    • Pass at least one exam
    • Pass both exams
    • Maximise the combined mark
  • The exact way the improvement is calculated matters: a fixed improvement of 5 percentage points vs an improvement of 5%