Explain this statistics/probability thing for me

[homer]

Sorry about the previous post but it does say Scholar under my name ..Doh

 

[GreedyAlgorithm]

Fair enough, but that becomes unquantifiable pretty fast because generally there are too many options for how to spend the dollar. All I'm looking at is the point at which playing the lottery is no longer a tax on people who are bad at math.

This is a very, very, very common misconception. Playing the lottery is often the CORRECT choice. The problem is when you look at the lottery as a game between two entities, the lotto runners and the lotto players. Then yes, it's simply a one-way transfer of money. But this is the fallacy of composition. You cannot evaluate the game in this way and conclude that for each individual playing the lottery is a bad idea. Why?

What you're really saying is that if an individual were to play the lottery enough times, he'd be losing money. Well, with the astronomical odds, no individual plays the lottery enough times. An example: say I offered you the chance to wager a penny. 99999999/100000000 of the time you lose your penny. Otherwise, you gain $900,000 dollars. If you played this game a billion times, you'd lose. If you played it once, you'd almost certainly lose what you might drop on the sidewalk and never know anyway, but you might get a huge windfall. This relates to the relative value of money. It is not linear. You can say it becomes 'unquantifiable pretty fast' but that doesn't absolve you of the responsibility to take it into account.

The lotto runners are playing a numbers game. For them you may use strict $$ expected value calculations. For the individual, you must use u($), the utility derived from the money instead, or you will draw erroneous conclusions. The individual must use money for it to be worth anything. Would you say that grocery stores are just a tax on those too dumb to realize they're giving away cold hard cash and not receiving anything in return? Lotto players are paying money for a chance to receive more money than they could possibly have gotten otherwise. They have decided that paying about $5,000 over a lifetime (for instance) is well worth the miniscule chance that at some point they'll be handed boatloads of cash.

Is this a bad decision? Maybe so. But it's certainly not clear-cut. And if you still think it is a clearly bad decision, consider the following lotto. I'm going to assume you make about $50,000 a year. Modify the problem according to your salary.

Once per year, you are allowed to bet the U.S. Government $40,000 dollars. No one else is allowed, just you, and only once each year. You get to write down the numbers 1 through 8 in some order, and so does the Government. If your order is the same or exactly the reverse of the Government's, you get $1,000,000,000 dollars. Otherwise, you lose your $40,000. What is the expected value of this bet? Nearly $10,000 in your favor. Would you make it? I wouldn't. Even though the expected value is in my favor.

Actually, I might save up over a lifetime and go for it in my last few years on the earth, 'cause that'd be awesome, but you get the point.

 

[Cabbage]

It's amazing that something (probability that is ) that has a range of between 0 and 1 can be so complicated . 0 means no chance at all , 1 means it absolutely will.

Actually, that's not true.

Say, for example, we're picking a real number at random from the closed interval [0,1]. There are various ways to model this, Lebesgue measure is a common, natural way to do this.

Since Lebsgue measure is uniform, the probability of picking any particular number is zero. On the other hand, some number will be picked, and this is an example of something that has probability zero, yet still happens.

Conversely, the probability of not picking a particular number is 1. Again, however, some number will be picked, and so here we have an example of something with probability one that doesn't happen.

 

[homer]

Actually, that's not true.

Say, for example, we're picking a real number at random from the closed interval [0,1]. There are various ways to model this, Lebesgue measure is a common, natural way to do this.

Since Lebsgue measure is uniform, the probability of picking any particular number is zero. On the other hand, some number will be picked, and this is an example of something that has probability zero, yet still happens.

Conversely, the probability of not picking a particular number is 1. Again, however, some number will be picked, and so here we have an example of something with probability one that doesn't happen.

Got in a bit of a muddle here .
I never claimed any values for chance or possible exceptions , merely stating what defines probability . So a coin has probability .5 of a head being thrown or .5 of a tail . You could argue about the possibility of it landing on it's side if you want ,I don't.
The only way you could get a probability of 1 that no particular number will be chosen would be if the sample size were infinite as would be the case for a random real number . However this number would be by its very nature undefined , since it would itself be infinitely long .
Playing with infinity leads to madness so I think I'll leave this discussion now .

 

[tsg]

The only way you could get a probability of 1 that no particular number will be chosen would be if the sample size were infinite as would be the case for a random real number . However this number would be by its very nature undefined , since it would itself be infinitely long .

This is true. But it has also left the realm of discrete probability.