I regularly use randomness to make (small) decisions in my life. I look at my watch and decide based on the last digit of the seconds. (Often, I just want one bit—even or odd.)
Without this, I'm indecisive all too often, in the stupidest of places. Choosing what to order at a restaurant, for example, used to be a bit of a pain; now I just select a few acceptable choices and then decide randomly. Similarly, it's also how I vary up my walk home: even, I take a left turn, odd—a right.
My parents used the last digit on a stopwatch to decide which of the kids got to pick the radio station in the car and similarly trivial decisions. (If 0-9 couldn't be divided evenly among the kids present, leftover digits would result in a re-roll.)
This trick takes away two types of pressure:
(1) trying to remember past decisions to be fair. Over a long enough period of time, with a fair coin/die/watch, everyone will get about the same chances. Or you'll walk all the routes about equally. Or you'll equally sample all of the acceptable menu items.
(2) trying to justify which decision was made. There's no need for explaining why Bob got to sit in the front seat, or why you ordered the chicken, or why you walked on 3rd street instead of 2nd. There's no need for second-guessing whether your decision accounted for all of the information and was really reasoned through correctly.
I saw the true value of removing the "justify every decision" step when a friend of mine was suffering from a severe mental illness that made it difficult to make trivial decisions. The result of trying to justify every decision: standing in the kitchen for several minutes because "I want to make a peanut butter sandwich, but I can't decide whether to get the bread or the knife first." By creating an automatic process for making decisions that are only slightly less trivial, we free ourselves to focus on the more important things in life.
> My parents used the last digit on a stopwatch to decide which of the kids got to pick the radio station in the car and similarly trivial decisions. (If 0-9 couldn't be divided evenly among the kids present, leftover digits would result in a re-roll.)
Easily solved by taking the [EDIT: seconds digits] modulo the number of children, but I guess that might be difficult while driving.
EDIT: People, please try to think this through before posting an innumerate reply.
Can you admit you were wrong yet? Unless the number of kids evenly divides 60, then some kids will have better odds than others. Just some small admission that the odds don't work out perfectly unless N is 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, or 60.
Edit: For instance, if there are 59 kids, the first one has twice as good a chance to win as any of the others, because both 0 % 59 and 59 % 59 are 0. In other words, in the set derived from x % 59, for x in {0, 1, ..., 59}, 0 appears twice, and each other number in the range 0-59 only appears once.
I never made the claim you're objecting to. The modulo operator minimizes the bias, it doesn't eliminate it. And it's the most efficient way to solve the problem.
My suggestion is that you stop thinking like a lawyer and start thinking like a scientist.
It's already solved. If there are 3 kids, 1-3 go to kid A, 4-6 to kid B, 7-9 to kid C, and 0 means try again. There's no bias present in this solution. Since it only takes a fraction of a second to generate a new digit with a stopwatch (using the hundredths of seconds digit) it's no big deal if you have to try five times.
Your solution introduces unnecessary arithmetic (what's 47 mod 3?) and unnecessary psychological pressure (will the kids start an argument while you're figuring it out? Will the ones who are better at math start taunting the others?) and it introduces bias in a family as large as ours got (7 or 8 kids can't evenly divide 60; 3, 6, or 7 kids can't evenly divide 100.)
A re-roll is simply a superior solution.
[Appeal to authority: my dad had a math degree and was valedictorian of his engineering college. I have bachelors and masters degrees in applied math. My oldest brother competed at the national level in Mathcounts. We all agree this is the best practical solution for this problem. Based on the votes, so does the bulk of the technically-minded Hacker News community.]
Another way to say it is that if a glance at the last clock digit turns out to be equal to to less than the number of children, it's accepted as the result, otherwise the operator looks away for a suitable interval and tries again.
> " tries again.... In other words, digit(s) modulo"
"Try again" and "modulo" do not mean the same thing.
Try again means you discard a result outside of the range, and then generate a new number without any reference to the value of the first try. If you have 7 kids and you're picking 0-9, you discard any results of 7-8-9 (or 8-9-0) and try again.
Modulo means you take a result outside of the range, divide it by the range, and use the remainder (which is guaranteed to be in the range) as your new answer. The problem with this approach is that it's not evenly distributed -- if you have 7 kids and digits 0-9, three of your kids will get double chances. Even if you use 0-59 as your range, some of the kids will have 9 chances and others will have 8 chances.
EDIT: It must be really, really satisfying to be able to anonymously downvote posts that actually reveal such things as grade-school arithmetic -- thus contributing to the intellectual wasteland that most accurately represents America in the early 21st century.
you're getting upset because you're making a terminology error, you say "digit" but mean "digits". A digit is either a finger, or the numbers 0-9, singular.
I'm getting outraged by the number of brainless cyberlawyers who think downvoting a post constitutes a creative act, on a par with knowing something about the topic of discussion.
The pattern is clear and ubiquitous -- if you actually know something, and if you're foolish enough to post a clear exposition, the morons who never post anything coherent will downvote your posts with a probability approaching certainty.
The modulo operator is the obvious solution to the original question. And if there were't an atmosphere of pervasive anti-intellectualism at HN, this thread would be much shorted than it is.
0 appears nine times.
1 appears nine times.
2 appears nine times.
3 appears nine times.
4 appears *eight* times.
5 appears *eight* times.
6 appears *eight* times.
This is not fair, and is an actual problem in real life: Two examples off the top of my head is when drawing cards in electronic poker games (where slight biases in the PRNG can be exploited by those who notice them) and in cryptography.
> Well, you are incorrect. The simple use of modulo does result in a bias.
You're arguing against something I never said. Large numbers reduce the bias, they don't eliminate it. Notice that earlier I considered the objection of a hypothetical clever child by using the minute digits as well. All these steps only minimize the bias in favor of small numbers, that cannot be eliminated. Even a 64-bit random number possesses this property. And the classic remedy for this bias is to apply the modulo operator to get the desired range.
And the classic remedy for eliminating bias is to redraw/reshuffle if the number is outside of the desired range. Note, this completely eliminates all bias. The negative aspect is that in this case, random number generation can take arbitrarily long (with rapidly diminishing probabilities); to make it more efficient, you could first use a modulo operation, but only using a base that divides the original range (e.g. for 7 kids and 60 seconds, do 60 mod 10, then redraw if the result is 7, 8 or 9).
You've failed to acknowledge the trade-offs of using your solution versus the OP's solution (or the solution above), and you're calling people who are trying to point out the trade-off "morons", that's why you're getting downvoted.
> to make it more efficient, you could first use a modulo operation, but only using a base that divides the original range (e.g. for 7 kids and 60 seconds, do 60 mod 10, then redraw if the result is 7, 8 or 9).
This makes it less efficient - in this case, 18/60 chance of having to redraw, versus 4/60. Taking the modulus of the largest range you can leaves less residue (what lutusp is alluding to, without realizing others are describing a simple way of getting an exact uniform distribution).
If you want to make it more efficient (in terms of entropy used), you need to save the 1-of-prime (eg bits, trits, etc) that have been successfully chosen (uniformly). For example, with 6 children and an 8 sided die (3 bits), a roll of 6 or 7 would narrow the choices to only children 0-2 or 3-5 respectively. Your subsequent rolls could then be done with a 4 sided die.
You're right, I don't know what I was thinking... Anyways, about your second paragraph, doesn't that only work if the number of desired outcomes (children) is non-prime?
>Would that solution be "multiply by the number of children and divide by ten"
No; his/her solution is: If 0-9 couldn't be divided evenly among the kids present, leftover digits would result in a re-roll, exactly as I said above.
> No, I am being downvoted because I'm right
Are you? This is math we're talking about, right and wrong are very precisely defined. For the sake of discussion, can you repeat the question you claim to have the right/correct solution to, and your answer?
The last digit on the clock can only be 0 through 9. If you truncate your chart accordingly, 0 and 1 are represented more than 2 and 3. Or do you mean “the entire seconds value” (for which this idea would work with four children) instead of “the last digit”?
It actually doesn't matter how many options are in your random number generator. You only get a normal distribution if the number of children you have divides the number of options in your generator. Even if you're using the seconds of a clock, it doesn't work for 7 children.
if the stopwatch uses base 10, the last digit will be in the range [0, 9], making n >=10 irrelevant. in this range, 0 and 1 appear 3 times; 2 and 3 appear 2 times.
if the stopwatch uses hex, then 0-3 will appear equally often in this case
The second digits on my digital watches extend to 59, and if a precocious youngster were to correctly object that a limit of 59 produces a bias, I would extend the range by multiplying the minute digit by 60 and adding it to the seconds (thus reducing the bias). We are, after all, most likely using a calculator to produce the result.
> if the stopwatch uses hex ...
I think we're safely in the red herring zone now. :)
EDIT: Yes, of course -- when confronted by a correct and reasoned reply, downvote it to discourage any such mistakes in the future.
i think you're confused on what a 'last digit' is. the seconds digits (plural) contain two digits, which range from 0-59; the last digit (singular) ranges from 0-9
> You want to multiply the last digit by the number of children and divide by ten.
That doesn't produce the result you think it does. Think a bit harder. If the last digit is 20, and there are four children, the result is 4 * 20 / 10 = 8. There is no eighth child.
Yeah, I was joking. The point is that either way you get a non-uniform distribution for n children unless n divides 10. I think the part you are misunderstanding in your responses is that the last digit of a clock has only ten options: 0 through 9. In the tables you have posted, children 0 and 1 each win 30% of the time, while children 2 and 3 each win only 20% of the time.
With a single generation, you can't uniformly choose from a set of n options with a random number generator that outputs m options, unless n divides m.
Do you actually think the point is to argue with people who actually know things, like a lawyer instead of a scientist? Using a modulo operator is the obvious solution to the original question, and cowardly anonymous downvotes can't erase simple facts -- but they eventually make them have so little contrast on the display that they become unreadable. What an achievement.
Nearly everytime you mention downvotes: the reason is tone and attitude. I don't know if you are aware just how much your tone influences the downvotes.
> Nearly everytime you mention downvotes: the reason is tone and attitude.
Look -- I posted the obvious solution to a common problem. Everything else depends on those who can't stand the unwelcome intrusion of simple, easily stated facts.
My "attitude" is that this problem is easily solved, using simple arithmetic. Imagine you're a scientist -- as such, do you object to a useful result because of its source?
When you get a downvote, ignore it. It's probably accidental or meaningless and these tend to self correct over a day.
When people downvote a bunch of your posts in a thread IGNORE THE DOWNVOTES. Vindictive downvote sprees are usually corrected over the course of the day. Mentioning the downvotes will usually prevent those corrective upvotes. Me tioning the downvotes in the unpleasant way that you do will attract downvotes.
You may wish to consider how you're presenting the information. If the first table of numbers gets downvotes posting the same table again without more information is going to get the same downvotes.
You could argue that it should not be this way, but it is.
> You could argue that it should not be this way, but it is.
Indeed it is. State facts, back up what you say with evidence, get downvoted. As certain as sunrise.
> If the first table of numbers gets downvotes posting the same table again without more information is going to get the same downvotes.
The alternative is to adopt the standards of religion instead of science. In religion, if you start losing followers, you change the mythology. In science, evidence is evidence, and how people feel about it has no standing.
> When you get a downvote, ignore it. It's probably accidental or meaningless ...
Easily proven false. My downvotes inevitably accompany anything I post that has evidence and/or links to references. If I offer an uncorroborated opinion or philosophical remark, however irrelevant or baseless, it's treated neutrally. This is how science works -- you observe things dispassionately and don't let yourself to be swayed by what people think is true.
Well, judging from your usual quality of comments (you're one of the users I remember for good contributions in science threads) I think there's an disagreement here that is not stated explicitly.
Do you disagree with any of the following statements, and if yes, where and why?
- original stopwatch solutions gives equal probability for any of the kid to get picked
- your modulo solution does not give equal probability for any of the kid to get picked
- you argue that the non-uniformness of your solution is not relevant in practice, because number of kids is likely to be an order of magnitude smaller than the readout from the stopwatch (i.e. last two digits)
> Do you disagree with any of the following statements, and if yes, where and why?
Okay. I usually expect people to locate the errors in their own thinking, but in this case, I'll make an exception. Here's one of the suggestions: "multiply the last digit by the number of children and divide by ten". Let's see how this works out:
The OP could have tested his suggestion before posting. I certainly would have.
I'm going to avoid your straw men, with this exception:
> you argue that the non-uniformness of your solution is not relevant in practice
I never said that. I said that the error could be minimized. But consider my test of the the alternative listed above. The error inherent with a relatively small number of children and a relatively large original number, say, 0 - 59, is a smaller error than the proposed alternative.
Hey, I asked you three simple questions with pretty much boolean answers, in order to try and clarify where exactly you end up disagreeing with everyone. Please be charitable.
> Here's one of the suggestions: "multiply the last digit by the number of children and divide by ten".
Hey, that one was clearly meant as a joke, and is not the suggestion I was referring to. If something is a strawman here, this is. The one I asked about is the "My parents used the last digit on a stopwatch to decide which of the kids got to pick the radio station in the car and similarly trivial decisions. (If 0-9 couldn't be divided evenly among the kids present, leftover digits would result in a re-roll.)".
Anyway, I must say I found the technique I used above quite effective at figuring out continous disagreements. You state some simple true/false statements describing your assumptions and ask the other party to agree/disagree and explain the points of disagreement. Apply recursively if needed. Kind of a discussion equivalent of git bisect ;).
> "I posted the obvious solution to a common problem"
You posted a solution that was less correct than the solution you were responding to.
The solution I originally posted is fair and practical. Divide the digits as evenly as you can between the kids, and treat any extras as a do-over. While it's technically possible to never end, in practical terms it produces a fair result within ten seconds. (The fact that you're trying to "fix" this solution confuses me; I wonder if perhaps you didn't understand it in the first place and have been arguing based on a misunderstanding ever since.) By contrast, your solution does not produce a fair result for certain numbers of kids. I promise, the kids in my family would have discovered that bias by the time there were 7 of us.
> "can't stand the unwelcome intrusion of simple, easily stated facts"
One of the reasons you've gotten a lot of downvotes here is because of the combination of:
- posting an incorrect solution
- insisting the incorrect solution is correct, in spite of several people showing you why it is not, and
- complaining that people who are downvoting are the ones in the wrong, including several insults (calling people airheads, innumerate, brainless cyberwarriors, saying they can't stand simple facts, etc.)
At times I really enjoy your comments. I've sent a lot of upvotes your way over the years. But in this case you're wrong, and you've taken to insulting the people who are patiently trying to explain to you why you're wrong. I know you're capable of better. Please take the time to reread and understand the original solution, and stop insulting people.
I'd like a reliably quantum source of random numbers to make certain decisions for me.
That way, if the many-worlds model is true, I can console myself that even if my decision turns out to be a bad one, at least many of my parallel-universe counterparts have made the other one.
In the days when Sussman was a novice, Minsky once
came to him as he sat hacking at the PDP-6.
"What are you doing?", asked Minsky.
"I am training a randomly wired neural net to play Tic-tac-toe", Sussman replied.
"Why is the net wired randomly?", asked Minsky.
"I do not want it to have any preconceptions of how to play", Sussman said.
Minsky then shut his eyes.
"Why do you close your eyes?" Sussman asked his teacher.
"So that the room will be empty."
At that moment, Sussman was enlightened.
I don't quite understand this.
A person's preconceptions is the collection of his previous knowledge about the subject.
What does it mean to have random preconceptions?
I think the point is that you can't remove external constraints from a situation simply by willing it (make the people in the room disappear, make your AI independent of assumptions).
For example the type of neural net chosen, the reinforcement and other parameters, the rules of the game chosen by the creator are all constraints which are not randomly chosen, so randomly choosing the starting state is not necessarily going to remove preconceptions built in to the model, and it is still choosing some starting state, just not an arbitrary one chosen by the creator.
I'm not sure it really applies to this article though, which is about a different problem - saving the time spent worrying about unimportant decisions or those where you don't have enough inputs and choosing to take a random path.
I agree but I wouldn't take it too literally. The coin flip isn't binding but it can be fascinatingly revealing about your own opinions.
I see it as a way to actualise the consequences of the choice and cut through layers of intellectual abstraction. In some respects its a tool to let you engage emotional thinking to help make better decisions. On the flip, it can suddenly trigger a feeling of loss and regret. Our fundamental beliefs can be strangely out of reach when we think too hard.
A similar tool is just explaining your decision to another. I can get a flush of emotion e.g. embarrassment or shame, that you don't get when you just cogitate alone. Pretty useful for tough design decisions e.g. midway through explaining a particularly clever idea I find myself apologising... its time to rethink things!
Also if both decisions make you hesitate, it suggests that costs/benefits of both of them cancel each other out, so you may as well stop hesitating and go with what the coin tells you.
Actually, the real point of flipping a coin is that during the brief lapse of time during which the piece is in the air, you will know what's the choice you prefer. The result doesn't matter.
This is what I've always done! I agree with what another poster said about how revealing it can be -- you suddenly see yourself in the position where the decision had actually gone a certain way and you have to deal with the consequences. People always look at me funny when I explain it to them, though.
I discovered this the other day when trying to find out what "drawing lots" actually meant in the Bible, another form of letting chance decide things. Apparently we don't know exactly what people were using back then.
The Republic of Venice developed that idea to a system of great complexity for electing the Doge, that included copious injections of randomness. There is an interesting analysis of this protocol in a paper by HP laboratories, available here (pdf): http://www.hpl.hp.com/techreports/2007/HPL-2007-28R1.pdf
If you know more than nothing, the best choice is often still random -- just with different weights applied to the various options, which I hear can be calculated with something called game theory.
Let's suppose, for instance, that two roads diverge in a yellow wood and you have reliable information that road A is the correct path with 90% probability and road B is the correct path with 10% probability. The normal person takes road A and will be right 90% of the time.
On the other hand, if you try to be probabilistic about it and follow road B with 10% probability to reflect its 10% probability of being the right road, then your chances of picking the right road are 0.90.9 + 0.10.1 = 82%.
Maybe there's a more complex situation where the weighted random decision making is a better idea, but I'm not seeing it right now.
Since my background is poker, I'll use an (overly simplified) example from there.
There are situations where you want to manipulate your range (set of hands) in a given spot so that your actions make up a certain distribution.
For example, if you want your range to be made up of 30% bluffs, you can figure out how often you get to that spot with a bluff and how often you get to that spot with a strong hand. Since you're always continuing with the strong hand, you can then know what percentage of the time you need to continue to bluff in order to accomplish this. Since it doesn't matter which of those hands you choose to bluff with, it generally makes sense to use randomness to decide in the moment.
I don't play poker, but from what I understand, the reason for including randomness in your decision is to throw your opponents off. I.e. You're playing against people who try to predict your behavior, so you throw in a random component to make it harder for them.
In decisions not involving active adversaries, deciding at random doesn't give you anything over a deterministic solution.
Nearly, but just to nitpick, you're not including randomness, you're actively making a choice to make the resulting pattern difficult to distinguish from randomness. So, sometimes you raise by a large amount because you have a strong hand. You don't want other players to be identify that you have a strong hand by the fact that you just offered a large raise, so you have to also bluff sometimes, raising when you really don't have the cards. Your actions are not random, but they are perceived as random by the other players.
It happens when you're competing. Suppose you're trying to find some buried treasure that you know at the end of one of the roads. But another tribe is trying to kill you, and will be waiting at the end of one of the roads. In that case your best strategy (and theirs) is to make the choice of which road to take probabilistically.
Isn't this covered by Cynefin - http://en.wikipedia.org/wiki/Cynefin? Action first in the chaotic quadrant corresponds to making a fast decision in the face of unpredictability.
I think I'm very indecisive, I have hard time ordering stuff at restaurants without holding up lines. I need to find a method of deciding soon. I'll try the coin trick.
I have that problem too. I usually narrow it down to two or three options and ask the server/cashier what I want. About 80% of the time I go with whatever s/he says; other times, simply hearing his/her recommendation makes me realize what my actual preference is.
I recommend this book. It's really entertaining and thought provoking. I remember carrying a die everywhere a few weeks after reading this book, in order to add some randomness to my decisions. Silly, but fun!
Without this, I'm indecisive all too often, in the stupidest of places. Choosing what to order at a restaurant, for example, used to be a bit of a pain; now I just select a few acceptable choices and then decide randomly. Similarly, it's also how I vary up my walk home: even, I take a left turn, odd—a right.
It's a surprisingly useful little trick!