What force controls probability?

[Dave1001]

Thomas, Dunstan: my toy example was just illustrate the idea of "low probability of great harm vs chance of moderate benefit" and how the words irrational and rational shouldn't be used to describe a decision based on these probabilities. But I'll admit that Dunstan's modified version of my game 2 is a nicer example.

CaptainManacles: You seem to be thinking of the laws of large numbers (and misstating/misinterpreting it, which is common).

I think "irrational" and "rational" could still be used within this context, and in fact, we implicitly make these type decisions all the time. That's why it's interesting to me that we're not adaptively optimized to make these decisions rationally.

I probably don't have a perfect idea of the math, but I do think with the right respective risk percentages, it's irrational not to risk losing both eyes for a gain of $5. Would one submit to a one in a googleplex risk of losing one's eyes for a 50% chance at winning $5? I certainly would, and I'd play that game over and over again for the rest of my life. Even if the marginal utility of extra $5 bills was low for me, but significantly greater than zero. I think it would be irrational not to.

We already play these games all the time. Do you leave your house to go to work or do errands? By leaving your house, your risk of losing both of your eyes probably increases slightly. But the moderate rewards you get for doing whatever day of work or errands makes it more than worth it for you.

But I believe I've read that when risks for great loss are somewhat greater than microscopic (and salient to the assessor, such as death in a plane crash is for many folks) our species shows a marked tendency for irrational decision making.

 

[Jorghnassen]

Buying the lottery is an irrational decision, especially if you keep doing it.

But, rationally, the best way to survive is to avoid great harm, so it is very reasonable to want to eliminate that risk as much as possible (it cannot of course, be completely eliminated), so picking lower gains while reducing a potential but non negligeable probability of great harm, is in fact, quite rational, if you value survival or your current state of comfort above moderate benefits. Now if you don't run errands and don't work, you will eventually die of starvation.

PS: I guess I won't be making money off organ trafficking. Damn...

 

[Dave1001]

Buying the lottery is an irrational decision, especially if you keep doing it.

yes. That's taking advantage of the greed element of irrational risk calculation. Hence a lot of people consider the lottery to be a regressive tax on folks who irrationally calculate risk/reward percentages. Of course, it wouldn't be irrational with the right prize size, odds of winning, and price of ticket. I think there are folks who consciously search out lotteries worldwide where those ratios actually work out to the ticket purchaser's benefit.

But, rationally, the best way to survive is to avoid great harm, so it is very reasonable to want to eliminate that risk as much as possible (it cannot of course, be completely eliminated), so picking lower gains while reducing a potential but non negligeable probability of great harm, is in fact, quite rational, if you value survival or your current state of comfort above moderate benefits.

Now if you don't run errands and don't work, you will eventually die of starvation.

PS: I guess I won't be making money off organ trafficking. Damn...

I think the element you're not considering here are the actual relative risk percentages. For example the risk of dying in a plane crash on a commercial airliner is extraordinarily low, but very salient for some people. And so they'll irrationally forsake the moderate gains that they'd have a good chance of picking up as a result of such an airplane flight because of the very low probably chance of a great loss in the form of dying in an airplane crash. They would still probably get on the plane at some point (perhaps if there was a 95% chance of receiving $2 million dollars at the end of the plane flight), but their risk calculation becomes irrationally distorted because of the saliency of the great harm (despite it's very low probability of occuring).

 

[andyandy]

Buying the lottery is an irrational decision, especially if you keep doing it.

what price hope?

 

[drkitten]

Buying the lottery is an irrational decision, especially if you keep doing it.

Only under very specialized and unrealistic assumptions. (One of the big problem with game theory is that no one remembers the simplifying assumptions that Johnny VN had to make to get it off the ground.)


JVN assumed that the game player was "risk-neutral"; that the mere act of playing a game did not give him any pleasure or discomfort. Look at how many people go to considerable time and expense merely to get to a casino to play games, and you'll see how unrealistic that assumption is/was.

He also assumed that the "value" placed on the prize by the player was directly proportional to the monetary sum involved. Again, this is ridiculous; people value dollars differently depending upon how many they have. That's why I no longer eat pot noodles; the "value" to me of eating food that doesn't taste like salty styrofoam is now worth more than the financial cost of real food -- quite a difference from my "bright college days."

If the value of a dollar to you is sufficiently small, and the value of possibly winning several million is sufficiently large, buying a lottery ticket is extremely rational.

 

[Dave1001]

Only under very specialized and unrealistic assumptions. (One of the big problem with game theory is that no one remembers the simplifying assumptions that Johnny VN had to make to get it off the ground.)


JVN assumed that the game player was "risk-neutral"; that the mere act of playing a game did not give him any pleasure or discomfort. Look at how many people go to considerable time and expense merely to get to a casino to play games, and you'll see how unrealistic that assumption is/was.

He also assumed that the "value" placed on the prize by the player was directly proportional to the monetary sum involved. Again, this is ridiculous; people value dollars differently depending upon how many they have. That's why I no longer eat pot noodles; the "value" to me of eating food that doesn't taste like salty styrofoam is now worth more than the financial cost of real food -- quite a difference from my "bright college days."

If the value of a dollar to you is sufficiently small, and the value of possibly winning several million is sufficiently large, buying a lottery ticket is extremely rational.

Sure, this is important in theory. And not hard to test empirically (which I'm sure has been done in great depth). However, in practice, the question is whether most lottery players have a good sense of what their actual odds to winning per dollar invested is. Do they accurately assess the odds add on an "entertainment factor" and then choose to buy the lottery ticket? Or do they have an inflated sense of their chances of winning? If they have an inflated sense of their chance of winning (due to the saliency of the 100 millions dollars or whatever), then I would posit that their decision making-process is irrational, in a way similar to a person who refuses to fly on a plane because of an inflated sense of the chance of death.

 

[drkitten]

However, in practice, the question is whether most lottery players have a good sense of what their actual odds to winning per dollar invested is.

No, that's an entirely irrelevant question.

They gain more pleasure from the thought that they might win a million dollars than they lose from not having a relatively nominal sum available for other spending.

Ergo, it's rational.

 

[Filip Sandor]

The idea of behind a "force" causing the 50/50 heads or tails landings is a misconception of a what is really just a mathematical model of the coin toss. In reality the coin simply does what it does - except that it can only land one of two ways - heads or tails... which gives you the "50/50" model used to help interpret the outcome, roughly, before you make a toss; it does not force the outcome however.

Take for example a game of roulette at the casino: you have roughly a 50/50 "chance" that the ball will land on either RED or BLACK (excluding the GREEN zeros). Let's assume your friend joined one of the tables before you came in the casino and has had a chance to already see 3 of the results and lets say all three were BLACK... in his mind, according to the 50/50 "chance model," the next hit is more likely be a RED than a BLACK, but for you, who has just joined the table now the chances are 50/50 even. Does this change the past 3 hits on the board? No, of course not. Does it affect the next hit? The answer is of course, no. Probability is a mathematical concept so it does not 'affect' reality. Quantum mechanical probability is based on certain physical calculations which are made prior to an outcome which is not the same thing because on the roulette wheel and the coin toss there are no real useful calculations made before the toss and therefore: no real accurate prediction can be made on what will come 'out' since you don't know what is going 'in'. Hope that helps.

 

[Kaarjuus]

I have only skimmed this thread because you are all much too clever for me I think it was Hawking who said that quantum theory is actually quite deterministic, and if we could measure what was happening with all the particles at the quantum level, we could predict anything at the macro level. But of course there are too many particles, so what we see appears to be randomness. For practical purposes, we can treat this pseudo-randomness as the real thing.

I doubt that Hawking said this, because it's patently untrue. The Heisenberg principle of quantum mechanics states that one cannot measure values (with arbitrary precision) of certain conjugate quantities, which are pairs of observables of a single elementary particle. One cannot with certainty find out both the position and the momentum of a particle.

 

[dogjones]

OK, so the outcome of one toss doesn't affect the next one, so therefore it is a 50/50 chance of either outcome at every toss, regardless of the next one. Fine.

In that case, how do you calculate odds of outcomes within a "set" of tosses? Suppose you take a coin with the intent of tossing it 10 times. Before you start the tossing, you calculate the odds of at least 1 toss of this set of 10 turning up heads. What would be the correct calculation?

After that, you begin tossing (oooo! hark at the double entendre!). The first toss comes up tails. The odds for "one of the tosses in this set of 10 coming up heads" now must change, because one of the outcomes is now known - yes? How?

You do this until the last toss. They have all been tails. I am assuming the odds for "one of the tosses in this set coming up heads" are now at 50/50?

Or were they always at 50/50, from the first toss?

 

[drkitten]

In that case, how do you calculate odds of outcomes within a "set" of tosses?

Er, you take a course in probability or discrete mathematics? Calculations like this can get tiresome rather quickly, and the calculations can be somewhat lengthy.

The basic principle is that if the probability of x is p_x, and the probability of y is p_y, then the probability of both x and y is p_x times p_y if the two events are independent. Similarly, if the probability of x is p_x, then the probability of not-x is 1 minus p_x.

Then you just work it out by cases.

Suppose you take a coin with the intent of tossing it 10 times. Before you start the tossing, you calculate the odds of at least 1 toss of this set of 10 turning up heads. What would be the correct calculation?

Well, the probability of any given coin coming up tails is 1/2. So the probability of two independent tosses coming up tails is 1/2 times 1/2, or 1/4. The probability of *three* independent tosses coming up tails is 1/2 times 1/2 times 1/2, or 1/8. Similarly, the probability of all ten tosses coming up tails is 1/2 times itself ten times, or 1/1024.

If not all tosses came up tails, then at least one must have come up heads. So the probability of at least one heads is 1- (1/1024) or 1023/1024.

After that, you begin tossing (oooo! hark at the double entendre!). The first toss comes up tails. The odds for "one of the tosses in this set of 10 coming up heads" now must change, because one of the outcomes is now known - yes? How?

After the first coin toss has been performed, the odds are no longer 50/50 -- "odds" really only describes events that haven't already happened. Otherwise, I'll give you 2:1 on the Pittsburgh Steelers winning the 2006 Superbowl. Any takers?

 

[Kaarjuus]

In that case, how do you calculate odds of outcomes within a "set" of tosses? Suppose you take a coin with the intent of tossing it 10 times. Before you start the tossing, you calculate the odds of at least 1 toss of this set of 10 turning up heads. What would be the correct calculation?

You add all the probabilities of all possible outcomes where at least one toss is heads together. The probability of the first toss being heads and all the rest tails, then the probability of the first toss being tails, the second toss being heads and the rest tails, and then... and so on until all the possibilities have been exhausted.

After that, you begin tossing (oooo! hark at the double entendre!). The first toss comes up tails. The odds for "one of the tosses in this set of 10 coming up heads" now must change, because one of the outcomes is now known - yes? How?

There are now not ten, but nine tosses. The sum of probabilities changes, as there are less numbers.

You do this until the last toss. They have all been tails. I am assuming the odds for "one of the tosses in this set coming up heads" are now at 50/50?

Yes, now the odds of getting heads are 50:50, which they have been for every single coin toss til this one.

 

[Dave1001]

No, that's an entirely irrelevant question.

They gain more pleasure from the thought that they might win a million dollars than they lose from not having a relatively nominal sum available for other spending.

Ergo, it's rational.

I think your own qualifiers make it a relevant question. For example "the thought that they might win" and "relatively nominal sum": that implies the assumption that they are accurately making just the type of calculations I was talking about. If they aren't, by implication of your own post, they're not making a rational choice.

I can understand a theory that the cost of the tickets are based solely on the marginal utility of that amount of money for most people. But if the odds of winning (beside their being greater than zero) didn't matter at all to the rationality of the choice than lottery companies should make the odds 1 in a googleplex. Of course the odds do matter. But are the typical ticket buyers assessing them rationally? In other words, most lottery ticket buyers may rationally be able to say that 1 in a googleplex odds for 1 million dollars in exchange for a $1 ticket is not rational enough odds for them to get pleasure in buying a ticket. But are they rationally analyzing odds that aren't so obviously extreme?



My previous post, for ease of reader comparison:

However, in practice, the question is whether most lottery players have a good sense of what their actual odds to winning per dollar invested is. Do they accurately assess the odds add on an "entertainment factor" and then choose to buy the lottery ticket? Or do they have an inflated sense of their chances of winning? If they have an inflated sense of their chance of winning (due to the saliency of the 100 millions dollars or whatever), then I would posit that their decision making-process is irrational, in a way similar to a person who refuses to fly on a plane because of an inflated sense of the chance of death.

 

[Molinaro]

Buying the lottery is an irrational decision, especially if you keep doing it.

In Canada, all lotteries are non-profit operations. The money either goes to charity, or to government coffers.

Therefore, buying a ticket is always a rational decision, because I'm 100% ok with where that money goes.

What I see you doing to defend your comment, quoted above, is that you are offering irrational reasons why someone might buy a ticket -- and then calling it irrational in general.

That doesn't come across as the least bit convincing.

 

[Meffy]

The force that controls probability is magnetism. Specifically, little electromagnets under the roulette wheel.

Y'know, I've got to find a less dishonest casino. =@.o=

 

[Dave1001]

In Canada, all lotteries are non-profit operations. The money either goes to charity, or to government coffers.

Therefore, buying a ticket is always a rational decision, because I'm 100% ok with where that money goes.

What I see you doing to defend your comment, quoted above, is that you are offering irrational reasons why someone might buy a ticket -- and then calling it irrational in general.

That doesn't come across as the least bit convincing.

I think he was mostly limited just by human language and efficiency, counting on us to see his points without endless wordy footnotes. I suspect loopholes of the type you described were already completely forseeable to him when he made his post, as they probably were to most of us. His general point, that buying lottery tickets is an irrational way to achieve $1 million net worth holds up. If one buy a lottery ticket for a million dollar lottery, one is usually buying something worth quite a bit less than the possibility of getting one million dollars in that time frame.

A good question is what is the most a probability expert who designed or runs the lottery would pay for the ticket . I assume that is likely to be the highest rational price. How would one value the odds of winning $1 million dollars? Is $1 the fair price for 1 in a million odds? Is $0.50 the fair price for 1:2million odds? Let's ignore the marginal utility of money, taxes, and the time value of money to start with. Is it that simple or are there more complicated factors that I'm missing?

 

[Filip Sandor]

OK, so the outcome of one toss doesn't affect the next one, so therefore it is a 50/50 chance of either outcome at every toss, regardless of the next one. Fine.

In that case, how do you calculate odds of outcomes within a "set" of tosses?

However you calculate it, it will only be a math model of what might happen and not what WILL happen. That's the whole point of probability theories - they depict possibilities not certainties. Trust me, I tried my luck at the casino with this probability crap when I was a bit younger and it doesn't work - the cards don't listen to the numbers in my head.

 

[Kaarjuus]

Therefore, buying a ticket is always a rational decision, because I'm 100% ok with where that money goes.

So, essentially, you buy lottery tickets to support charities or government coffers. Which is, of course, a good cause.

But why not give the money to the charities directly? So that there's no possibility of you winning the lottery and therefore depriving the charities of the money you and everyone else gave them by buying tickets? If the point is to give money to the charities, why risk them losing the money?

 

[69dodge]

Take for example a game of roulette at the casino: you have roughly a 50/50 "chance" that the ball will land on either RED or BLACK (excluding the GREEN zeros). Let's assume your friend joined one of the tables before you came in the casino and has had a chance to already see 3 of the results and lets say all three were BLACK... in his mind, according to the 50/50 "chance model," the next hit is more likely be a RED than a BLACK,

Why would he think that?

If I were him, I wouldn't think that.

 

[69dodge]

No, that's an entirely irrelevant question.

They gain more pleasure from the thought that they might win a million dollars than they lose from not having a relatively nominal sum available for other spending.

Ergo, it's rational.

I don't think I understand you.

It seems like you'd say that anything anyone does is rational, because, after all, they decided to do it, didn't they? But I guess you don't mean that.

Can you give an example of what you'd consider irrational behavior?