1. ## How do you think about the odds?

Since many people here are students, I will start with a simple example, relevant to testing.

Suppose you are taking a test that subtracts points for wrong answers and gives points for correct answers. Say there are 5 answers for each question, and you loose a fifth of a point for a wrong answer and gain a point for a correct answer. Pretty standard, the pay-off of plain guessing is 0. But, if you eliminate an answer, the expected value improves to 0.4. Does that mean you guess every time you can eliminate an answer?

Suppose the scenario had more details. Suppose there is only one question on the test that you need to guess on and you can eliminate only one answer. Suppose further that the score on this test is a once in a life-time thing and that being a high scorer would give you something that you really desired. Keep in mind that the most likely result is a lower score, and you are more likely to score lower than higher because of that guess.

OK. So the above example is a rather contrived. But I wanted to find out how people evaluated risk/reward.

Let's up the stakes (What follows is also contrived, but I wanted to see how people think about this).

Suppose X is your entire life savings at the age of 50. You are now given an challenge to guess a random (uniformly distributed) number between 1 and 10. If you guess right, you will get 100*X. If you guess wrong, you will loose X. Do you take the challenge?

The expected value is 9.1X. That is, on average, you will gain a little over 9 times your life savings. If scientists did a study of people who accepted the challenge versus people who did not, those who accepted would be about 10 time more rich on average (all else being equal). Sounds good, huh?

The probability of loosing is 90%. That is, 9 out of 10 people who took the challenge would go broke. If scientists did a study of people who accepted the challenge versus people who did not, those who accepted would be much more likely to be broke (all else being equal).

So would you take this challenge?

2. um, no. Risk is unquestionably much higher than reward in this scenario, and the downside (losing all your retirement funds at age 50) is far worse than the upside (becoming rich at age 50) as well as far more likely. This illustrates why median is often more relevant than mean.

You might have to go with a more grey situation if you want more interesting answers.

3. Yeah. I guess, I wanted to see the contrast, because believe it or not, I've met people who say taking the challenge is the "correct" answer, no matter how you "feel".

But let's tone it down a bit.

Instead of the challenge being one guess at a random number (uniformly distributed) between 1 and 10 inclusive. Your challenge is to guess at 10 random numbers (independent, uniformly distributed) between 1 and 10 inclusive. For each correct guess, you get 10*X, for each incorrect guess you get -0.1X. (X is still your life savings at age 50)

The expectation value is still 9.1X. However, the distribution is different.

Would you take the challenge now?

Does knowing the distribution above, change your mind?

4. Expected return doesn't matter so much given sufficiently small sample sizes, random uncontrollable variables, and especially given high standard deviations. What you have described is, essentially, gambling. There's nothing wrong with gambling provided that a) you have the risk tolerance to make a rational decision and b) that you can afford to lose.

My experience is that the scenario described (low probability significant upside with high probability of moderate disaster) isn't so common, but its inverse (high probability moderate upside with low probability significant disaster) is. That is, people tend to ignore low probability but highly damaging events, like natural disasters and economic collapses.

5. Originally Posted by dala
Expected return doesn't matter so much given sufficiently small sample sizes, random uncontrollable variables, and especially given high standard deviations. What you have described is, essentially, gambling. There's nothing wrong with gambling provided that a) you have the risk tolerance to make a rational decision and b) that you can afford to lose.

My experience is that the scenario described (low probability significant upside with high probability of moderate disaster) isn't so common, but its inverse (high probability moderate upside with low probability significant disaster) is. That is, people tend to ignore low probability but highly damaging events, like natural disasters and economic collapses.
You may want to check the second scenario, then. All, I did was spread out the risk in the first scenario over 10 events. I am interested what happens when the risk of ruin (though still very large) is not more likely than the probability of high rewards.

6. My thoughts are the same for both scenarios. Both have small sample sizes, random uncontrollable variables, and high standard deviations, and I would consider both gambles. I would be more likely to take the bet in the second scenario, but my criteria for taking it would remain the same (sufficiently high risk tolerance and ability to swallow the loss).

7. Regarding the first, I'd take the risk.

Regarding the second and third..

The problem with most games and decisions (in game and decision theory) is that they largely require several iterations (along the lines of your third example, sort of) in order to reach a more agreeable equilibrium. The 'rational' decision is not always the 'best' one. In addition, if we're factoring in marginal utility (&etc.), each additional dollar is not worth as much as the last.

Long and short, there is a point at which the risk of losing x far outweighs the potential benefit of gaining 100*x. More like.. let me risk 10% of x on that gamble and keep 90% in a CD, or something..

I'm ambitious, but I'm not balls-out 'let's lose everything' ambitious. In that specific scenario, I'd probably have enough in retirement to live a comfortable life that it would not be worth the risk of me losing it all, regardless of whether the odds say that I can win much more. Then again, I'm not one who places value on having a lot of money, either. (I do, however, value being able to eat.)

One of my philosophies is that life is a game of playing the odds and that we cannot be absolutely certain of absolutely everything; there is always some risk involved.

I'd have to think more about how to reconcile all of this. Neat subject

8. Originally Posted by ygolo
Let's up the stakes (What follows is also contrived, but I wanted to see how people think about this).

Suppose X is your entire life savings at the age of 50. You are now given an challenge to guess a random (uniformly distributed) number between 1 and 10. If you guess right, you will get 100*X. If you guess wrong, you will loose X. Do you take the challenge?

The expected value is 9.1X. That is, on average, you will gain a little over 9 times your life savings. If scientists did a study of people who accepted the challenge versus people who did not, those who accepted would be about 10 time more rich on average (all else being equal). Sounds good, huh?

The probability of loosing is 90%. That is, 9 out of 10 people who took the challenge would go broke. If scientists did a study of people who accepted the challenge versus people who did not, those who accepted would be much more likely to be broke (all else being equal).

So would you take this challenge?
90% chance of losing what I've worked my whole life for? Never.
Originally Posted by ygolo
Yeah. I guess, I wanted to see the contrast, because believe it or not, I've met people who say taking the challenge is the "correct" answer, no matter how you "feel".

But let's tone it down a bit.

Instead of the challenge being one guess at a random number (uniformly distributed) between 1 and 10 inclusive. Your challenge is to guess at 10 random numbers (independent, uniformly distributed) between 1 and 10 inclusive. For each correct guess, you get 10*X, for each incorrect guess you get -0.1X. (X is still your life savings at age 50)

The expectation value is still 9.1X. However, the distribution is different.

Would you take the challenge now?
No.
Does knowing the distribution above, change your mind?
No. But make it -0,01x/*1x and I'd probably risk a decent ammount of cash.

9. @ygolo Let me test you too. Did you think you were gonna walk away untested?

Suppose you had 100 billion dollars. You are now given this challenge:
A person is gonna pick a number between 1 and 10 and write it in a paper. He's gonna tell you that if you get it right you lose everything, but if you get it wrong you get to be a trillionaire (and I'm not talking about somalian shillings).
Do you take the challenge?

10. Math is best used as a tool applied to life, not so much life applied to math. Mathematically, the odds are in favor of taking the chance. 1 out of 10 times you will win 100 times X. (.1*100x)-(.9X)>X. A practical calculation would be different than this though, where any decision not resulting in at the bare minimum 50% or greater chance of success is considered unfavorable odds.

You can say they're the same ratio in different sizes, but they aren't when you're considering a closed range of what X is. One example of this is poker bankroll management... the general rule of thumb is to risk only about 1/20th of your total pool of funds available in play at any given time. This is because in no-limit games (where, like your scenario, you may risk all your chips) you can bet all your money at the early stages of the hand when the odds are in your favor and still lose. The small sample size ensures enough breathing room for your statistically favorable decisions to pay off.

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