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%

Outreach: Article Competition 2011

Projection Point is an Irish company providing risk analysis, consulting and behavioural testing. As part of its website, Projection Point maintains a blog on observations about risk intelligence and comments on risk stupidity.

Our Outreach Programme holds an annual essay competition, the winner of which will walk away with €200 and have their essay published on our blog.

For the 2011 Essay Competition, we invite secondary school students, undergraduate and postgraduate university students to submit essays on the topic of risk. The exact topic of choice is up to the author, but the final essay should fit in nicely within the area of risk intelligence and behavioural science.

So what are we looking for? Dylan’s C4ISR article acts as a good example.

Rules

    1. All essays must be original; no previously published material will be accepted.
    2. Each participant can submit a maximum of two essays.
    3. Quotes and references must be clearly marked throughout the essay and properly cited. It is the author’s responsibility to ensure that all quoted material (including statistics) is correct, relevant, and accurate. The judges will verify your citations and in case of purposely misleading or erroneous facts and citations, might disqualify the essay.
    4. Any form of plagiarism will result in automatic disqualification.

Every participant must provide the following personal details*:

    • Name and Surname, Valid Postal Address, and Email Address.
  1. Projection Point reserves the right to publish the essay on projectionpoint.com and/or any subdomains (such as blog.projectionpoint.com).
  2. The decision of the Jury is final and is not subject to appeal.
  3. Copyrights of winning entry will apply to Projection Point.
  4. Projection Point is held responsible for ensuring the author’s name is kept and publicized with the essay. It also reserves the right to remove/delete the essay without notice to or consent by the author.
  5. Upon request by the author, Projection Point must publish essays anonymously.
  6. There is no limit on the contestant’s age or occupation. Everyone is welcome.

Guidelines

  • Entries should be no shorter than 1000 words. However, we would like to emphasize that quality is more important than quantity.
  • You may include images in your essay
  • Although not mandatory, we suggest all entries Harvard System of Referencing
  • Your essay should include the following:
    1. An introduction, which introduces the topic and contains an explanation of your position.
    2. A body, which contains the main content of your essay.
    3. A conclusion, which summarizes the subject, research, and analysis presented in the essay and sets forth your conclusions.Your essay should also include notes and/or a bibliography except when using ‘APA style’.
  • Reference notes (footnotes or endnotes) give the sources of your information or ideas. Footnotes are placed at the end of the essay.
  • A bibliography is a list of the works that you have referred to in your essay or have consulted in order to write it. Essays are encouraged to use a variety of sources—academic journals, news magazines, newspapers, books, government documents, publications from research organizations, etc.
  • The Internet or World Wide Web should not be the only source for your essay. Be aware that you may encounter “republished” or “third generation” information on the Internet that is inaccurate or improperly attributed. When citing Internet sources, you must include the following information: author(s), title of work, Internet address, and date information was accessed. Detailed instructions can be obtained from the manuals listed above. For the purposes of this essay, Internet sources should be listed separately from non-electronic sources, such as books, magazines, and newspapers.

Good Essays

  • Should take up at least two and a half A4 pages.
  • Are well-written.
  • Demonstrate that you have researched/investigated the topic.
  • Are submitted in plain text, HTML, PDF or MS-Word format.
  • Stay focused on the topic.
  • Use quotations and references.

Avoid

  • Spelling and grammatical errors.
  • Approaching the topic so that it can only be understood by someone who is already an expert in the field.
  • Omit references to backup statements.
  • Submit essays that overlap heavily with articles previously published on Projection Point.

Prizes

First Prize: The winning entry will receive a cheque of €200.
Second Prize: Contestant will receive a cheque of €75.
Third Prize: Contestant will receive a cheque of €50.
Fourth – Tenth: Contestant will receive three vouchers for the Projection Point Expert RQ Test.

A selection of the top ten essays will be published on our blog (blog.projectionpoint.com).

Winning highschool / secondary school students writing on behalf of their school make their academic institution’s library eligible for a free copy of “Risk Intelligence: How to Live with Uncertainty“.

Review Process
Every submission will be read by at least three reviewers and judged against six equally weighted criteria: Thematic focus, Skill in argumentation, Clarity of ideas, Originality, Precision and Style in writing, and Grace of expression.

For each criterion there is an associated scale from 0 to 10. Each essay’s review score will be calculated as follows: the average of the six equally weighted criteria for each reviewer is calculated. Thereafter the average of the reviewers score will be calculated. In addition to the score, each reviewer will add a final comment for the author. The highest scoring essays will win.

Winners will be announced before or on January 31st 2011.

Submission Deadline: December 20th 2011.
Submission Details: Entries may be sent to submission@blog.projectionpoint.com or be posted to:
Projection Point Ltd
Building 1000, Second Floor,
City Gate, Mahon,
Little Island,
Cork, Ireland.

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* Winners may choose to have their essay published anonymously