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%

Risk Intelligence for the Layperson

If you intend to embark on a career as a gambler or a weather forecaster, or if you intend to speculate on the stock exchange or take a punt at your local horse races, improving your risk intelligence would stand you in good stead and improve your odds of winning. However, not everybody is interested in gambling or other ‘risky’ ventures, so why should you care about your risk intelligence?

In everyday situations, improved risk intelligence can be advantageous. Whether it be evaluating what insurance cover we might need, if we are a member of a jury, making business investment decisions, working in recruitment and employing someone with previous convictions and even whether or not to move in with a girlfriend/boyfriend, the ability to accurately assess the risks of each circumstance would leave us much more likely to get a positive final outcome. The layperson in particular exhibits three key flaws when evaluating risk:

1) Overconfidence – this is the most common flaw. It is when we are more confident in a particular outcome than we have reason or evidence to back it up with. Overconfidence is exceedingly common and often comes from those that we trust most to help us make decisions; experts. Relying on ‘experts’ is not always as simple as it might sound. Just because someone knows a lot about something, doesn’t mean they are aware of their limitations, of how much they don’t know. For evaluating risk, it is often more important to know what you don’t know, than it is to know what you do know. Therefore, by improving your own risk intelligence, you can evaluate how effective an expert is, how reliable their information is, and what risks you are taking to apply it to your own situation. The sort of experts that you would meet on a regular basis that would fall into this category would be doctors and other medical staff, and bank managers and other financial advisory staff.

2) Worst-Case Scenario Thinking – This is when we allow ourselves to be negative and make low-probability scenarios into near certainties, without any logic to back it up. It often creates with it a spiral of demoralising thoughts and even-worse outcomes, none of which are helpful for making a decision. It prevents genuine risk assessment and logical reasoning for a decision and often results in an emotive and poor decision. An example of this could be refusing medical treatment such as surgery to fix a badly-broken leg because of the risk of a low-probability outcome such as death. It can be very easy to become irrational and emotionally-involved and decide that opting for surgery will result in death and therefore it would be better to just live with a bad limp or inability to walk, the outcome of leaving the leg to heal by itself.

3) The Availability Heuristic: Imagination Inflation – estimating the likelihood of something happening based on how easily we remember it happening in the past. The illogicality of this can be seen by the fact that what we are remembering doesn’t need to have happened in real life, but we could have seen it in a film or a computer game. This can result in grossly inflated probabilities. It can also work the other way, with difficult to visualise scenarios being underestimated. For example, we could have been watching the film The Towering Inferno and the following week we come to renew our business’s fire insurance. Vividly remembering the burning building from that film, you are more likely to spend unnecessary money on more comprehensive and expensive cover than you actually require.

By being aware of these flaws and evaluating your thought processes against them when assessing risky decisions you can improve your risk intelligence and the general success of any risk-based decisions you make. Assessing how sure you are something is right or wrong, or is or isn’t going to happen, rather than simply looking at things from a black or white perspective as well as evaluating the outcomes of any decisions you take to improve on your thought process in the future will help you on your way to becoming a more astute risk taker.

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Jessie Hardcastle is a freelance writer from England who specialises in finance and investment for a number of UK journals and blogs. With a growing following she has recently be focussing mainly on the problems close to her London home as Europe continues to falter in the face of political indecision.

The unbearable lightness of expected utility

In chapter 8 of my book, Risk Intelligence, I discuss the concept of expected utility.  To calculate the expected utility of a course of action, the first step is to estimate – separately – the probability of success, and the potential gains and losses that success and failure would entail.  Next, one does a little math, multiplying the probability of success by the potential gains, and multiplying the probability of failure by the potential losses.  Finally, we adding these two figures together to end up with the expected utility of that course of action.  After doing this for each possible action, the rational choice is to pick the action with the highest expected utility.

Expected utility is an abstract concept. It doesn’t refer to any actual win or loss: it’s the average amount you would win or lose per bet if you placed the same bet an infinite number of times. But this ethereal figure takes on an almost physical nature for the expert gambler, looming even larger in his consciousness than the actual profit or loss that hypnotises the rest of us. Expert gamblers see something different when they look at a poker table or a roulette wheel. Most people see a range of prizes; they see a single abstract figure.

This was brought home to me one night when a highly successful gambler invited me to accompany him to a casino. This particular gambler made all his money betting on horses. He shunned casinos, and only ventured in on this occasion because he wanted to give me and his other companions the thrill of seeing someone playing for high stakes. We followed him over to the craps table.

Craps is a dice game in which players place wagers on the outcome of the roll of two dice. My gambler friend proceeded to show off by placing bets of 1000 US dollars on the pass line, one after the other. A pass line bet is won immediately if the first roll is a 7 or 11. If the first roll is a 2, 3 or 12, the bet loses (this is known as “crapping out”). If the roll is any other value, it establishes a point; if that point is rolled again before a seven, the bet wins. If, with a point established, a seven is rolled before the point is re-rolled, the bet loses (“seven out”). A pass line win pays even money; in other words, my friend stood to win or lose a thousand dollars on each bet.

However, as he told me later over cocktails in the casino bar, it wasn’t this figure that was at the forefront of his mind. Instead, he just treated each bet as a bit of fun that cost him fourteen bucks. Although not a casino gambler, he was familiar enough with craps to know that the expected value of a pass line bet is -0.014. And this meant that, on average, each bet of a thousand dollars would leave him fourteen dollars less well off. The actual profit or loss at the end of an hour on the craps table could be anything from plus fifty thousand dollars to minus fifty thousand, and my friend would be aware of this figure too. But the figure that mattered most for him was the expected value of minus fourteen bucks per bet. That’s not a good-value bet, of course. Which is why, in any other circumstance, my gambler friend would never play craps.