Two or three months ago, my trip to Las Vegas made me ponder the following: If all gambles in the casinos have negative expected values, why do people still engage in gambling - especially my friends fairly well-versed in probability/statistics?
Suffice it to say, I still have not answered that question.
On the other hand, this did lead me to ponder more about whether rational behavior always involves making choices with the highest expected (or positive) value - call this Rationality-Expectation (R-E) hypothesis.
Here I'd like to offer some counterexamples that show R-E is clearly false, to me at least. (In hindsight, these look fairly trivial but some commentators on this site speak as if maximizing expectation is somehow constitutive of rational decision making - as I used to. So, it may be interesting for those people at the very least.)
- Suppose someone offers you a (single trial) gamble A in which you stand to gain 100k dollars with probability 0.99 and stand to lose 100M dollars with probability 0.01. Even though expectation is -98999000 dollars, you should still take the gamble since the probability of winning on a single trial is very high - 0.99 to be exact.
- Suppose someone offers you a (single trial) gamble B in which you stand to lose 100k dollars with probability 0.99 and stand to gain 100M dollars with probability 0.01. Even though expectation is 98999000 dollars, you should not take the gamble since the probability of losing on a single trial is very high - 0.99 to be exact.
A is a gamble that shows that choices with negative expectation can sometimes lead to net pay off.
B is a gamble that shows that choices with positive expectation can sometimes lead to net costs.
As I'm sure you've all noticed, expectation is only meaningful in decision-making when the number of trials in question can be large (or more precisely, large enough relative to the variance of the random variable in question). This, I think, in essence is another way of looking at Weak Law of Large Numbers.
In general, most (all? few?) statistical concepts make sense only when we have trials numerous enough relative to the variance of the quantities in question.
This makes me ponder a deeper question, nonetheless.
Does it make sense to speak of probabilities only when you have numerous enough trials? Can we speak of probabilities for singular, non-repeating events?