# 100% chance at $1.08M or 97% at $36M? **URL:** \ **Category:** In My Humble Opinion\ **Created:** [November 22, 2017, 2:03pm UTC](https://boards.straightdope.com/t/100-chance-at-1-08m-or-97-at-36m/802402 "2017-11-22T14:03:31Z")\ **Posts on this page:** 20\ **Page:** 6
**Author:** ![MoonMoon](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/moonmoon/32/2959_2.png) [@MoonMoon](https://boards.straightdope.com/u/MoonMoon)\ **Post date:** [November 24, 2017, 1:19am UTC](https://boards.straightdope.com/t/100-chance-at-1-08m-or-97-at-36m/802402/101 "2017-11-24T01:19:40Z")
> [@JPrescott](#): > > I’m living in poverty. Even $30k would be life changing for my family. No WAY would I risk losing. Same. It seems to me that most people who’d go for the bigger money are saying they’re already set for retirement. I might actually be looking at eating catfood in 20 years, so I’m taking the million. ---
**Author:** ![Corry\_El](https://avatars.discourse-cdn.com/v4/letter/c/ecae2f/32.png) [@Corry\_El](https://boards.straightdope.com/u/Corry_El)\ **Post date:** [November 24, 2017, 2:22am UTC](https://boards.straightdope.com/t/100-chance-at-1-08m-or-97-at-36m/802402/102 "2017-11-24T02:22:58Z")
> [@Machine\_Elf](#): > > Expected value, by itself, is not enough to make the decision. Instead of a 97% chance of winning $36M, suppose it’s a 9.7% chance of winning $360M. Or a 0.97% chance of winning $3.6B. Those all have the same expected value, but (using the last case) do you really want to risk the $1.08M in your hand for a 99.3% chance of walking away with nothing? Sure. Modern finance for decades has dealt in term of return and standard deviation of return. Those two measures also don’t necessarily explain _everything_, a lot of apparent (professional financial) market anomalies are arguably because they don’t. But the two together provide much more guidance than just return. For example the expected return of 3 zeroes and 97 $36mil wins every 100 tries is around $6mil. For 97 zeroes and 3 $1.2bils, which has a slightly higher expected return, the std dev is around $205mil. For 1 zero and 99 $35.27mils, again about the same expected value, the std dev is around $3mil. So bringing in the second measure quantifies the intuition that super big payoff at low probability for same expected return is a worse deal. $6mil is much closer to zero, no risk. But mean-variance analysis doesn’t tell you what risk is appropriate for each person, that differs. Again though seems to me some people hypothetically turning down a highly favorable deal such as proposed in some cases must be make real world decisions at odds with that. For example I guess some people saying they might turn down almost sure (97% qualifies in my book) $36mil because retirement is not well set are investing (real existing money) in stocks\* to try to get retirement shored up. I wonder how unlikely they think it is the stock market (or markets they invest in) pulls a Japan for the next few decades? Or worse. I don’t know the probability of that but it’s not necessarily less than 3%. And that’s v an expected return of maybe high single digits per year. Here somebody is offering a 36:1 return instantly, with a 3% chance of meltdown, then you can invest in very low risk investments forever after and still be way ahead (for a long time probably anyway) of where you were investing 1mil in stocks. It doesn’t compute to turn that down, to me, unless you need money _now_ or very bad things will happen. \*which they might be doing anyway even if they don’t want to, through a pension plan that might go belly up if the stock market experiences long term poor performance. ---
**Author:** ![Velocity](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/velocity/32/18006_2.png) [@Velocity](https://boards.straightdope.com/u/Velocity)\ **Post date:** [November 24, 2017, 2:40am UTC](https://boards.straightdope.com/t/100-chance-at-1-08m-or-97-at-36m/802402/103 "2017-11-24T02:40:12Z")
I wonder if Dopers are an unusually cautious breed. I’d guess that if this poll were put to the U.S. population at large, it would be \*overwhelmingly \*in favor of the 97%, mega-bucks option. ---
**Author:** ![Tamerlane](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/tamerlane/32/7602_2.png) [@Tamerlane](https://boards.straightdope.com/u/Tamerlane)\ **Post date:** [November 24, 2017, 2:54am UTC](https://boards.straightdope.com/t/100-chance-at-1-08m-or-97-at-36m/802402/104 "2017-11-24T02:54:48Z")
> [@Corry\_El](#): > > For example I guess some people saying they might turn down almost sure (97% qualifies in my book) $36mil because retirement is not well set are investing (real existing money) in stocks\* to try to get retirement shored up. Has absolutely zero to do with that for me :). I already have sufficient retirement income set aside for every aspect of retirement except some quite specific uncertainty over housing at SF Bay Area prices. The average attached home price out this way is now a cool [$905k](http://www.newgeography.com/content/005796-san-francisco-bay-area-housing-3q17-affordability-limits-continue-be-tested-average-prices-hit-record-levels), with existing stocks below $500k at 12%. Most of _those_ are less desirable for a variety of reasons. [Rents](http://www.mercurynews.com/2017/07/03/bay-area-rents-see-summer-jump/) meanwhile continue to creep up to keep pace. $1 million dollars in hand means a fully paid-off average house in a nice neighborhood( neither a mansion nor a “great” neighborhood )and I get to stay in an area I’m fond of. No $$ if I hit that 3% roll means some uncertainty over whether I continue to live at all in the area after retirement or just have to make do with a less desirable spot. I don’t need or desire to be _rich_ - guaranteed comfortable is more than fine. ---
**Author:** ![Machine\_Elf](https://avatars.discourse-cdn.com/v4/letter/m/82dd89/32.png) [@Machine\_Elf](https://boards.straightdope.com/u/Machine_Elf)\ **Post date:** [November 24, 2017, 4:03am UTC](https://boards.straightdope.com/t/100-chance-at-1-08m-or-97-at-36m/802402/105 "2017-11-24T04:03:03Z")
> [@Voyager](#): > > But you can’t compare against yesterday when you didn’t have the money. You have to compare to the time right after you got handed the money. Here in our hypothetical world you might not think this is relevant, but the [Endowment Effect](https://en.wikipedia.org/wiki/Endowment_effect) says that chances are your opinion will be different after you are handed the money. The money in your hand will be worth over a million to you, effectively. (Maybe not enough for you to not take the bet.) And I suspect you would be in a worse state of mind if you lost the bet than if the test had never happened. > > Point is, the moment you get the money it is not the house’s money, it is yours. Good salespeople know this, and will try to get their merchandise physically into your hands if at all possible. Happened to my wife and me a few years ago at a jewelry shop, where a salesperson saw us eyeballing a diamond in the display case, and insisted on taking it out and handing it to us for closer inspection. I knew what was happening, and kind of chuckled to myself. It also figured into The Bourne Identity, in [the scene where Jason first meets Marie.](https://www.youtube.com/watch?v=HSckms_rfwY&t=0m34s) He tosses her $10,000 and asks for a ride to Paris; when she declines, he asks for his money back, and now suddenly she’s thinking about having to give back the $10,000 she’s actually holding in her hands. She takes the deal. 😃 > [@Corry El](#): > > For example I guess some people saying they might turn down almost sure (97% qualifies in my book) $36mil because retirement is not well set are investing (real existing money) in stocks\* to try to get retirement shored up. I wonder how unlikely they think it is the stock market (or markets they invest in) pulls a Japan for the next few decades? Or worse. Stocks come with risk, but safer investments (e.g. bonds) are pretty much guaranteed to generate an inadequate nest egg for retirement. It’s pretty rare to find publications that advise young and middle-aged people to put more than a small fraction of their retirement investments in bonds. Once people _do_ have a large enough nest egg for retirement (or become dependent on their nest egg, as by actually retiring), they are generally advised to dial back the stocks so as to reduce their risk. ---
**Author:** ![Mijin](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/mijin/32/9369_2.png) [@Mijin](https://boards.straightdope.com/u/Mijin)\ **Post date:** [November 24, 2017, 4:40am UTC](https://boards.straightdope.com/t/100-chance-at-1-08m-or-97-at-36m/802402/106 "2017-11-24T04:40:54Z")
This is probably fighting the hypothetical, but in any weird situation where I’m getting free cash, my tendency would be to take the simpler option. If I pick $1m, and there’s not money in my pocket at the end of the transaction, I know I’ve being cheated. ---
**Author:** ![Voyager](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/voyager/32/133_2.png) [@Voyager](https://boards.straightdope.com/u/Voyager)\ **Post date:** [November 24, 2017, 6:22am UTC](https://boards.straightdope.com/t/100-chance-at-1-08m-or-97-at-36m/802402/107 "2017-11-24T06:22:28Z")
> [@Machine\_Elf](#): > > I think the relevant concept here is [opportunity cost:](https://en.wikipedia.org/wiki/Opportunity_cost) in order to play the roulette wheel for a shot at $36M, you are sacrificing the opportunity to walk away with a certain $1.08M. People who rationalize trying for $36M on the basis that they will be no worse off than the were initially are ignoring the opportunity cost of their decision. That’s true. I was going with the psychological aspect. > [@](#): > > Expected value, by itself, is not enough to make the decision. Instead of a 97% chance of winning $36M, suppose it’s a 9.7% chance of winning $360M. Or a 0.97% chance of winning $3.6B. Those all have the same expected value, but (using the last case) do you really want to risk the $1.08M in your hand for a 99.3% chance of walking away with nothing? I think we’ve seen in the answers that utility varies with the situation of people, and also it does not grow linearly with the amount of money. For some of us the utility of winning $36 million is more than 36X that of winning $1 million. In others the utility of winning the $1 million is very great. I think for most people the utility of 3.6 billion is not much different from $360 million, and so is not worth the increased risk. I think expected value considering utility would give a more accurate result as to how people respond. ---
**Author:** ![actualliberalnotoneofthose](https://avatars.discourse-cdn.com/v4/letter/a/e79b87/32.png) [@actualliberalnotoneofthose](https://boards.straightdope.com/u/actualliberalnotoneofthose)\ **Post date:** [November 24, 2017, 8:31am UTC](https://boards.straightdope.com/t/100-chance-at-1-08m-or-97-at-36m/802402/108 "2017-11-24T08:31:32Z")
At this point in my life the marginal benefit of $1 million would be too much to pass up. It would mean not needing to work physical labor 12+ hours a day trying to feed my kids while managing to pay only a couple of bills a month late and wondering how I’m ever going to pay my student loans. ---
**Author:** ![TokyoBayer](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/tokyobayer/32/13989_2.png) [@TokyoBayer](https://boards.straightdope.com/u/TokyoBayer)\ **Post date:** [November 24, 2017, 2:00pm UTC](https://boards.straightdope.com/t/100-chance-at-1-08m-or-97-at-36m/802402/109 "2017-11-24T14:00:18Z")
> [@TokyoBayer](#): > > Still thinking about it so I haven’t answered the poll. Ran it past my wife. She’d take the safe money. I’m still not sure what I’d do. ---
**Author:** ![Suburban\_Plankton](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/suburban_plankton/32/219_2.png) [@Suburban\_Plankton](https://boards.straightdope.com/u/Suburban_Plankton)\ **Post date:** [November 24, 2017, 2:44pm UTC](https://boards.straightdope.com/t/100-chance-at-1-08m-or-97-at-36m/802402/110 "2017-11-24T14:44:32Z")
If it were a question of $100K vs $3.6 million, I’d roll the dice (err…you know what I mean). But I can’t walk away from a guaranteed million bucks. It wouldn’t turn me into Jed Clampett, but it would be a “life changing” sum. ---
**Author:** ![LSLGuy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lslguy/32/5813_2.png) [@LSLGuy](https://boards.straightdope.com/u/LSLGuy)\ **Post date:** [November 24, 2017, 3:15pm UTC](https://boards.straightdope.com/t/100-chance-at-1-08m-or-97-at-36m/802402/111 "2017-11-24T15:15:35Z")
ETA: Oops. Thought I was at the end of the thread. I’m not. So this is not as context-aware a post as I’d intended. :smack: Anyhow, ref expected value … Expected value is NOT a property of a single trial. Expected value is NOT a property of a small set of trials. It’s a property ONLY of a collection of trials sufficient to reveal the long run. Further … As several people have pointed out above, the length of the “long run” is directly connected to the likelihoods of the various outcomes. Informally stated, the long run is however many trials it takes for the \*least \*likely outcome to occur several times. Said another way, the actual total outcome of a trial set will converge to the naïve EV calc value at a rate related to how many times the least likely outcome has occurred. This asymptotic convergence is not linear. Even on a 50/50 proposition like a fair coin flip, EV is misleading in a single trial. Let’s say we bet $1 on the prop: “heads I win. Tails you win.” You and I can both compute an EV of $0.50. Guess what. That outcome has precisely zero chance of occurring. What will occur is $1 for me or $1 for you. That is all. Over 100 trials it’ll come out sorta close to $50 for me and $50 for you. Over 10,000 trials it’ll be even closer to $5,000 each. But even at 10,000 trials of a 50/50 proposition we’re only beginning to converge towards the EV. ---
**Author:** ![Velocity](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/velocity/32/18006_2.png) [@Velocity](https://boards.straightdope.com/u/Velocity)\ **Post date:** [November 24, 2017, 3:21pm UTC](https://boards.straightdope.com/t/100-chance-at-1-08m-or-97-at-36m/802402/112 "2017-11-24T15:21:54Z")
So far, people have twice referenced their wives going for the safe $1m; I wonder if women are more risk averse on this than men? ---
**Author:** ![Corry\_El](https://avatars.discourse-cdn.com/v4/letter/c/ecae2f/32.png) [@Corry\_El](https://boards.straightdope.com/u/Corry_El)\ **Post date:** [November 24, 2017, 3:29pm UTC](https://boards.straightdope.com/t/100-chance-at-1-08m-or-97-at-36m/802402/113 "2017-11-24T15:29:48Z")
> [@Machine\_Elf](#): > > Stocks come with risk, but safer investments (e.g. bonds) are pretty much guaranteed to generate an inadequate nest egg for retirement. It’s pretty rare to find publications that advise young and middle-aged people to put more than a small fraction of their retirement investments in bonds. Once people _do_ have a large enough nest egg for retirement (or become dependent on their nest egg, as by actually retiring), they are generally advised to dial back the stocks so as to reduce their risk. OK, but I don’t see how that necessarily answers my point. I’m sure some people who would take the $1mil are older and already largely set, ie they don’t plan to take much investment risk in real life, so why take any on such a hypothetical bet? They are just _very_ risk averse. Or in the limit someone might be relatively set, relatively old to enjoy money and have no heirs (or anyway not interested in what happens to money of theirs after they die). IOW they aren’t necessarily extremely risk averse, $36mil just isn’t worth much to them. But, if someone is younger and would be advised to put a high % of investments in stocks, then I see a potential outright inconsistency to say they wouldn’t go for the $36mil. Their real life money could largely disappear in a stock market meltdown _with no bounce back_, at least for decades. Some people tell themselves that won’t happen, I’m not saying advisers lie to people to make them think that, but tend to be content to let people think that. But it’s not so historically in the big picture. ‘The stock market always comes back’ is a US phenomenon of the American Century. It hasn’t always been true elsewhere in the past, there’s no gtee of it anywhere in the future. So I mean to say, if people correctly understood that, it would be IMO inconsistent to say ‘sure I have my 401k in stocks for the long haul, [so called]\* riskless investments won’t earn enough’, then shy away from a 3% chance of the $36mil bet going wrong. Maybe it’s just a very skewed population toward the first explanation to get an amazing, IMO, 40-ish % turning down the almost sure $36mil, or maybe there’s some inconsistency. I don’t know. \*The debt of highly indebted rich countries might eventually be subject to ‘haircuts’ those countries’ polities will choose to impose on ‘rich’ bondholders rather than generate runaway inflation to ‘just print the money to pay off the bonds’, the supposed fail safe that some people incorrectly equate to ‘riskless’. Those bonds are low risk relative to the stock market (or virtually any other investment) not literally riskless. ---
**Author:** ![puzzlegal](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/puzzlegal/32/3827_2.png) [@puzzlegal](https://boards.straightdope.com/u/puzzlegal)\ **Post date:** [November 24, 2017, 3:31pm UTC](https://boards.straightdope.com/t/100-chance-at-1-08m-or-97-at-36m/802402/114 "2017-11-24T15:31:04Z")
> [@Voyager](#): > > This is a great example of the economic irrationality Thaler and others have pointed out. Your 100% chance of getting $1.08 million is exactly equivalent to someone handing you the money and then asking you if you want to bet it in order to win $36 million. > > That you and **kopek** and I bet others thinks it is different shows that are putting the $1.08 million in a different mental bucket from money you think is yours. > > Consider this - would your perception be different if instead of “winning” the $1.08 million you just found out you inherited it, and are given the opportunity to bet it for $36 million. It is exactly the same situation, but it feels different, doesn’t it? And I think it actually is different, because: > [@Mijin](#): > > This is probably fighting the hypothetical, but in any weird situation where I’m getting free cash, my tendency would be to take the simpler option. > If I pick $1m, and there’s not money in my pocket at the end of the transaction, I know I’ve being cheated. In real life, nothing is ever a sure thing. There always might be a catch of some sort. And I think we are wired to take that into account. I said that if I truly believed this situation, I’d take the gamble with the huge payout. But if something like this actually happened to me, I would put an enormous “wtf is going on here?” discount on that gamble, and I might well walk away with the million. ---
**Author:** ![LSLGuy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lslguy/32/5813_2.png) [@LSLGuy](https://boards.straightdope.com/u/LSLGuy)\ **Post date:** [November 24, 2017, 3:33pm UTC](https://boards.straightdope.com/t/100-chance-at-1-08m-or-97-at-36m/802402/115 "2017-11-24T15:33:18Z")
> [@Corry\_El](#): > > Sure. Modern finance for decades has dealt in term of return and standard deviation of return. Those two measures also don’t necessarily explain _everything_, a lot of apparent (professional financial) market anomalies are arguably because they don’t. But the two together provide much more guidance than just return. > … > For example I guess some people saying they might turn down almost sure (97% qualifies in my book) $36mil because retirement is not well set are investing (real existing money) in stocks\* to try to get retirement shored up. I wonder how unlikely they think it is the stock market (or markets they invest in) pulls a Japan for the next few decades? Or worse. I don’t know the probability of that but it’s not necessarily less than 3%. And that’s v an expected return of maybe high single digits per year. … Excellent. There’s EV, there’s variance, and then there’s individual marginal utility. Those make up the mostly rational part of the decision. Then there’s the irrational stuff. Some decision makers are better at ignoring the irrational factors. Others are not. As to your second section about stock market investments I agree, but IMO there’s an additional factor. To a crude approximation, the broad stock market is a barometer of the economic health of the world. And like a weather barometer, it doesn’t control the outcome, it merely reacts to it. Like it or not, each of us is embedded in the world economy. It’s substantially impossible to insulate yourself from the ride, be it smooth or bumpy. No matter how much cash, gold, or ammo you stockpile, there’s still a possibility of a blacker swan than you expected that will overwhelm your defenses. Defenses that are costly to maintain until used. Said another way, there’s no opportunity to diversify away the fact you’re an Earthling. So I (or you) invest in the market not because it’s intrinsically good, but because it’s intrinsically inevitable. The existence of your house, your job, your country’s government, all that stuff is closely coupled to itself and the stock market is simply a (jittery) barometer of that larger reality. ---
**Author:** ![ThisIsTheEnd](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thisistheend/32/4713_2.png) [@ThisIsTheEnd](https://boards.straightdope.com/u/ThisIsTheEnd)\ **Post date:** [November 24, 2017, 3:49pm UTC](https://boards.straightdope.com/t/100-chance-at-1-08m-or-97-at-36m/802402/116 "2017-11-24T15:49:24Z")
Take the million. Variance is a bitch. ---
**Author:** ![Section\_Maker\_Jupe](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/section_maker_jupe/32/1424_2.png) [@Section\_Maker\_Jupe](https://boards.straightdope.com/u/Section_Maker_Jupe)\ **Post date:** [November 24, 2017, 4:33pm UTC](https://boards.straightdope.com/t/100-chance-at-1-08m-or-97-at-36m/802402/117 "2017-11-24T16:33:56Z")
Although Im risk averse, 97% seems like a “decent” chance…get the odds down to 80%, i start to think a bit harder about taking the 1 million instead of the gamble…get the odds to 50/50? and I would take the 1 million. ---
**Author:** ![Voyager](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/voyager/32/133_2.png) [@Voyager](https://boards.straightdope.com/u/Voyager)\ **Post date:** [November 24, 2017, 6:30pm UTC](https://boards.straightdope.com/t/100-chance-at-1-08m-or-97-at-36m/802402/118 "2017-11-24T18:30:52Z")
> [@LSLGuy](#): > > ETA: Oops. Thought I was at the end of the thread. I’m not. So this is not as context-aware a post as I’d intended. :smack: > > Anyhow, ref expected value … > > Expected value is NOT a property of a single trial. Expected value is NOT a property of a small set of trials. It’s a property ONLY of a collection of trials sufficient to reveal the long run. That is true - but it is only relevant if you do not know either the range of outcomes or probability of outcomes until you do a long trial. In the case in the OP you know both these things. So we know the expected values in both cases. Here it is used as a guide as to whether the bet is a good one or not. > [@](#): > > Further … As several people have pointed out above, the length of the “long run” is directly connected to the likelihoods of the various outcomes. Informally stated, the long run is however many trials it takes for the \*least \*likely outcome to occur several times. > > Said another way, the actual total outcome of a trial set will converge to the naïvehttp://boards.straightdope.com/sdmb/showthread.php?p=20620353#post20620353 EV calc value at a rate related to how many times the least likely outcome has occurred. This asymptotic convergence is not linear. You can use sampling theory to determine how close you are. While you may not know the probability of low outcome results, there are conservative measures which can be useful. > [@](#): > > Even on a 50/50 proposition like a fair coin flip, EV is misleading in a single trial. Let’s say we bet $1 on the prop: “heads I win. Tails you win.” You and I can both compute an EV of $0.50. Guess what. That outcome has precisely zero chance of occurring. What will occur is $1 for me or $1 for you. That is all. This has nothing to do with how EV is used. A ten cent bet with a dollar return is a good one since the EV is 50 cents. A 70 cent bet is a bad one. That is just as true for n = 1 as it is for n = 1,000,000. ---
**Author:** ![Doctor\_Jackson](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/doctor_jackson/32/32_2.png) [@Doctor\_Jackson](https://boards.straightdope.com/u/Doctor_Jackson)\ **Post date:** [November 24, 2017, 7:28pm UTC](https://boards.straightdope.com/t/100-chance-at-1-08m-or-97-at-36m/802402/119 "2017-11-24T19:28:35Z")
At 97% odds I would go for the $36MM. One million dollars would cut my estimated retirement time from 10 years to 5, $36MM cuts it to now. I like the sound of now! ---
**Author:** ![LSLGuy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lslguy/32/5813_2.png) [@LSLGuy](https://boards.straightdope.com/u/LSLGuy)\ **Post date:** [November 24, 2017, 9:22pm UTC](https://boards.straightdope.com/t/100-chance-at-1-08m-or-97-at-36m/802402/120 "2017-11-24T21:22:41Z")
Now that’s a concrete demonstration of the time value of money.🙂 [Previous page](https://boards.straightdope.com/t/100-chance-at-1-08m-or-97-at-36m/802402.md?page=5) [Next page](https://boards.straightdope.com/t/100-chance-at-1-08m-or-97-at-36m/802402.md?page=7)