# A Question About Probabilities (Lottery Related) **URL:** \ **Category:** Factual Questions\ **Created:** [August 25, 2011, 11:01pm UTC](https://boards.straightdope.com/t/a-question-about-probabilities-lottery-related/594096 "2011-08-25T23:01:11Z")\ **Posts on this page:** 15\ **Page:** 2
**Author:** ![md2000](https://avatars.discourse-cdn.com/v4/letter/m/73ab20/32.png) [@md2000](https://boards.straightdope.com/u/md2000)\ **Post date:** [August 27, 2011, 7:40pm UTC](https://boards.straightdope.com/t/a-question-about-probabilities-lottery-related/594096/21 "2011-08-27T19:40:27Z")
The first ticket gives you odds of about 1 in 200M; the second, about 1 in 100M. The third, 1 in 66M; the 4th, 1 in 50M; the fifth, 1 in 40M. The sixth about 1 in 33M. and so on… The law of diminishing returns - at a certain point the extra dollar is hardly worth the benefit. Buying that first ticket boosts your odds of winning dramatically… I think of lotteries as a stupidity tax - the dumber you are, the more you pay. I only have to pay $5/week. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [August 27, 2011, 8:35pm UTC](https://boards.straightdope.com/t/a-question-about-probabilities-lottery-related/594096/22 "2011-08-27T20:35:19Z")
> [@Indistinguishable](#): > > (That is, the marginal utility of a dollar needn’t _always_ decrease with income) Find me a person who acts like they have a non-concave utility function, and I’ll pay you a lot of money. Edit: That is to say, while it’s perfectly mathematically reasonable to write down utility functions like that, such a theory just doesn’t correspond to reality. ---
**Author:** ![gonzomax](https://avatars.discourse-cdn.com/v4/letter/g/e8c25b/32.png) [@gonzomax](https://boards.straightdope.com/u/gonzomax)\ **Post date:** [August 27, 2011, 8:55pm UTC](https://boards.straightdope.com/t/a-question-about-probabilities-lottery-related/594096/23 "2011-08-27T20:55:19Z")
years ago I calculated the Michigan Lotto chances. If you play the weekly football pool with 100 squares and it is 12 inches square, you would have a string of them over 19 miles long. One square would win in the Lotto. Pretty graphic representation. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [August 28, 2011, 3:51am UTC](https://boards.straightdope.com/t/a-question-about-probabilities-lottery-related/594096/24 "2011-08-28T03:51:09Z")
> [@ultrafilter](#): > > Find me a person who acts like they have a non-concave utility function, and I’ll pay you a lot of money. Sure. Go find someone who plays the lottery. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [August 28, 2011, 4:02am UTC](https://boards.straightdope.com/t/a-question-about-probabilities-lottery-related/594096/25 "2011-08-28T04:02:39Z")
That is, the theory that people make decisions consistent with expected utility theory under the assumption of decreasing marginal utility of money does not correspond to the reality that many people play the lottery and feel the particular distribution of uncertainty it offers them is preferable to the more certain results of not playing the lottery. If one is dogmatic about the theory, then this is by fiat a problem with people’s preferences (“You’re wrong to want this instead of that!”), but to one less dogmatic, it may just as well be considered a problem with the theory. ---
**Author:** ![BigT](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/bigt/32/12044_2.png) [@BigT](https://boards.straightdope.com/u/BigT)\ **Post date:** [August 28, 2011, 9:48am UTC](https://boards.straightdope.com/t/a-question-about-probabilities-lottery-related/594096/26 "2011-08-28T09:48:28Z")
> [@Indistinguishable](#): > > Sure. Go find someone who plays the lottery. When you actually talk to said people, you usually find out they think they have a greater chance of winning than they do. If they are working under incorrect information, then you can hardly say that that means the utility function is incorrect. So, instead, you have to pick things for which they do have and understanding. In other words, you have to pick some other behavior other than playing the lottery. ---
**Author:** ![brocks](https://avatars.discourse-cdn.com/v4/letter/b/7feea3/32.png) [@brocks](https://boards.straightdope.com/u/brocks)\ **Post date:** [August 28, 2011, 10:42am UTC](https://boards.straightdope.com/t/a-question-about-probabilities-lottery-related/594096/27 "2011-08-28T10:42:15Z")
> [@Buck\_Godot](#): > > That said, my wife sometimes buys lottery tickets saying she gets more than $1 worth of hopes and dreams out of them, and who am I to argue that. Evidently you are not the fulfillment of all her hopes and dreams. By the way, my wife doesn’t buy lottery tickets, thank you for asking 🙂 ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [August 28, 2011, 2:24pm UTC](https://boards.straightdope.com/t/a-question-about-probabilities-lottery-related/594096/28 "2011-08-28T14:24:04Z")
> [@BigT](#): > > In other words, you have to pick some other behavior other than playing the lottery. Or at least one person who understands the odds and has purchased at least one lottery ticket in their life (considering the potential winnings worth the cost) nonetheless. I’ve witnessed such things happen (or, at least, I think I have). ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [August 29, 2011, 2:44am UTC](https://boards.straightdope.com/t/a-question-about-probabilities-lottery-related/594096/29 "2011-08-29T02:44:14Z")
Eh, but whatever. I don’t know why I’m making such an argument out of it. Consider the matter dropped. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [August 29, 2011, 2:47am UTC](https://boards.straightdope.com/t/a-question-about-probabilities-lottery-related/594096/30 "2011-08-29T02:47:01Z")
For what it’s worth, [isoelastic utility](https://secure.wikimedia.org/wikipedia/en/wiki/Isoelastic_utility) with \eta slightly less than 2 gives a pretty good approximation to how people behave, at least in the studies I’ve seen ---
**Author:** ![septimus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/septimus/32/410_2.png) [@septimus](https://boards.straightdope.com/u/septimus)\ **Post date:** [August 29, 2011, 4:04am UTC](https://boards.straightdope.com/t/a-question-about-probabilities-lottery-related/594096/31 "2011-08-29T04:04:40Z")
> [@ultrafilter](#): > > Find me a person who acts like they have a non-concave utility function, and I’ll pay you a lot of money. I understand your point (and thanks for the very interesting link to _isoelastic utility_), however when [the man in the coon-skin cap in the big pen wants eleven dollar bills and you only got ten](http://www.youtube.com/watch?v=d_ujAXxNxU0#t=28s), some people would be happy to stake $10 against $1 on a coin flip, and be confident that this action was rational. ---
**Author:** ![septimus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/septimus/32/410_2.png) [@septimus](https://boards.straightdope.com/u/septimus)\ **Post date:** [August 29, 2011, 9:00am UTC](https://boards.straightdope.com/t/a-question-about-probabilities-lottery-related/594096/32 "2011-08-29T09:00:30Z")
> [@ultrafilter](#): > > For what it’s worth, [isoelastic utility](https://secure.wikimedia.org/wikipedia/en/wiki/Isoelastic_utility) with \eta slightly less than 2 gives a pretty good approximation to how people behave, at least in the studies I’ve seen How much would you pay for the chance to get 2^N dollars, where N is the number of consecutive heads you toss with a fair coin? This is the famous St. Petersburg Gamble, sometimes considered intractable, but utility functions make it workable. (Unlimited winnings are counterfactual, but the values shown below are almost identical with those assuming that Warren Buffet is banking the game and capping your winnings at 20 gigadollars.) For fun, I tried to determine St. Petersburg’s value with Isoutility: Eta = 1 With bankroll of $10, you should pay up to $4.7 With bankroll of $100, you should pay up to $7.0 With bankroll of $1000, you should pay up to $10.0 With bankroll of $10000, you should pay up to $13.2 With bankroll of $100000, you should pay up to $16.5 Eta = 2 With bankroll of $10, you should pay up to $3.5 With bankroll of $100, you should pay up to $5.6 With bankroll of $1000, you should pay up to $8.6 With bankroll of $10000, you should pay up to $11.8 With bankroll of $100000, you should pay up to $15.1 Eta = 5 With bankroll of $10, you should pay up to $2.4 With bankroll of $100, you should pay up to $4.3 With bankroll of $1000, you should pay up to $7.1 With bankroll of $10000, you should pay up to $10.2 With bankroll of $100000, you should pay up to $13.5 (Anyone care to verify these figures? :dubious: ) ---
**Author:** ![septimus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/septimus/32/410_2.png) [@septimus](https://boards.straightdope.com/u/septimus)\ **Post date:** [August 29, 2011, 10:43am UTC](https://boards.straightdope.com/t/a-question-about-probabilities-lottery-related/594096/33 "2011-08-29T10:43:48Z")
> [@septimus](#): > > Eta = 1 > With bankroll of $10, you should pay up to $4.7 > Eta = 2 > With bankroll of $10, you should pay up to $3.5 > Eta = 5 > With bankroll of $10, you should pay up to $2.4 After posting I realized I made a stupid off-by-X error in the calculations, with big effect only on the $10 bankrolls. These can be adjusted without redoing calculation to give: Eta = 1 With bankroll of $19.5, you should pay up to $4.7 Eta = 2 With bankroll of $17, you should pay up to $3.5 Eta = 5 With bankroll of $15, you should pay up to $2.4 (Still no guarantee, but the numbers are probably less wrong than before. ☹ Although again it will have to be “my word against yours”, my careless-error rate has gone up a lot since I joined SDMB. I don’t know whether to blame senility or apathy. :rolleyes: ) ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [August 29, 2011, 3:01pm UTC](https://boards.straightdope.com/t/a-question-about-probabilities-lottery-related/594096/34 "2011-08-29T15:01:55Z")
> [@septimus](#): > > I understand your point (and thanks for the very interesting link to _isoelastic utility_), however when [the man in the coon-skin cap in the big pen wants eleven dollar bills and you only got ten](http://www.youtube.com/watch?v=d_ujAXxNxU0#t=28s), some people would be happy to stake $10 against $1 on a coin flip, and be confident that this action was rational. Not paying the man presumably has some significant negative utility attached to it. ---
**Author:** ![CurtC](https://avatars.discourse-cdn.com/v4/letter/c/ce73a5/32.png) [@CurtC](https://boards.straightdope.com/u/CurtC)\ **Post date:** [August 29, 2011, 3:22pm UTC](https://boards.straightdope.com/t/a-question-about-probabilities-lottery-related/594096/35 "2011-08-29T15:22:52Z")
> [@Lare](#): > > :smack: Is there nothing The Google can’t do? It can also easily handle calculations involving units, and conversions. Try googling this: 100 km / 3 hours, in furlongs per minute [Previous page](https://boards.straightdope.com/t/a-question-about-probabilities-lottery-related/594096.md?page=1)