I have often seen discussions of what actions to take in the context of rare events in terms of expected value. For example, if a lottery has a 1 in 100 million chance of winning, and delivers a positive expected profit, then one "should" buy that lottery ticket. Or, in a an asteroid has a 1 in 1 billion chance of hitting the Earth and thereby extinguishing all human life, then one "should" take the trouble to destroy that asteroid.


This type of reasoning troubles me.


Typically, the justification for considering expected value is based on the Law of Large Numbers, namely, if one repeatedly experiences events of this type, then with high probability the average profit will be close to the expected profit. Hence expected profit would be a good criterion for decisions about common events. However, for rare events, this type of reasoning is not valid. For example, the number of lottery tickets I will buy in my lifetime is far below the asymptotic regime of the law of large numbers.


Is there any justification for using expected value alone as a criterion in these types of rare events?


EDIT: As many have pointed out, the article in Slate discusses many issues, and it is not fair to say that this article subscribes to this point of view. However, there are other sources which do appear to subscribe to it.



Expected value as decision criterion in the context of rare events
The [Kelly criterion](http://en.wikipedia.org/wiki/Kelly_criterion) is *the* optimal betting strategy for a player with limited resources (and if one had an infinite amount of capital they probably would not be interested in buying lottery tickets anyway). 


A Kelly player bets a fractional amount of their capital $X$ on each trial as to maximize the expected value of $\log X$. The strategy does not depend on the number of trials so it does not matter if the negative (or positive) outcome is a rare event. 


The Kelly criterion has several nice properties.


* Maximizing $\mathbb E[\log X]$ asymptotically maximizes the rate of capital growth.
* The expected time to reach a preassigned target capital is asymptotically smallest with a strategy maximizing $\mathbb E[\log X]$.
* The Kelly bettor is never behind any other bettor on average in $N$ trials.
* The Kelly bettor never risks ruin (and the bettor maximizing $\mathbb E X$ does; the variance can ruin them).


In practice, the Kelly strategy rules out buying lottery tickets for most rational investors (the amount of capital they can afford to bet is much smaller than the price of a typical lottery ticket). 


You might be interested in the papers ["Optimal Gambling Systems for Favorable Games"](http://www.bjmath.com/bjmath/breiman/breiman.pdf) by L. Breiman and ["The Kelly Criterion in Blackjack, Sports Betting, and the Stock Market"](http://www.pitt.edu/~sorc/trade/files/RiskManagement/kelly.pdf) by E. Thorp. There are also several nice articles on the subject by W. Ziemba in *Wilmott Magazine*.


