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what happens on average when a thousand people do something once is not a clue to what will happen when one person does something a thousand times. In this, it seems, evolved human instinct may be a much better at statistics than modern economists.fn4 To explain this distinction using an extreme analogy, if you offered ten people PS10m to play Russian roulette once, two or three people might be interested, but no one would accept PS100m to play ten times in a row.

It's important to recognize, both in lotteries and medicine, you can do things that change a probability by a large amount but with no real-world, practical significance. You can increase the odds of winning that state lottery by a factor of 100 by buying 100 lottery tickets. But the chance of winning remains so incredibly low, 1 in 100,000, that it hardly seems like a reasonable investment. You might read that the probability of getting a disease is reduced by 50% if you accept a particular treatment. But if you only had a 1 in 10,000 chance of getting it anyway, it may not be worth the expense, or the potential side effects, to lower the risk.

it is not how likely an event is to happen that matters, it is how much is made when it happens that should be the consideration. How frequent the profit is irrelevant; it is the magnitude of the outcome that counts. It is a pure accounting fact that, aside from the commentators, very few people take home a check linked to how often they are right or wrong. What they get is a profit or loss.

When we hear things like "There is a sixty percent chance that the conflict between these two countries will escalate to war" or "There is a ten percent probability that a rogue nation will detonate an atomic device in the next ten years," these are not calculated probabilities of the first kind; they are subjective expressions of the second kind, about how confident the speaker is that the event will occur. Events of this second kind are not replicable like the events of the first kind. And they're not calculable or countable like playing cards or fires on Elm Street. We don't have a bunch of identical rogue nations with identical atomic devices to observe to establish a count. In these cases, a pundit or educated observer is making a guess when they talk about "probability," but it is not a probability in the mathematical sense. Competent observers may well disagree about this kind of probability, which speaks to their subjectivity.

markets (and life) are not simple win/lose types of situations, as the cost of the losses can be markedly different from that of the wins. Maximizing the probability of winning does not lead to maximizing the expectation from the game when one's strategy may include skewness, i.e., a small chance of large loss and a large chance of a small win. If you engaged in a Russian roulette-type strategy with a low probability of large loss, one that bankrupts you every several years, you are likely to show up as the winner in almost all samples--except in the year when you are dead.

How could people miss such a point? Why do they confuse probability and expectation, that is, probability and probability times the payoff?

Pierre-Simon Laplace wrote in his 1812 book Theorie Analytique des Probabilites: "The most important questions of life are indeed, for the most part, really only problems of probability."

let's imagine four people toss the coin just twice. There are four possible outcomes: HH, HT, TH or TT, all of equal likelihood. So let's imagine that each of the four people starts with $100 and throws a different combination of heads and tails: HH HT TH TT The returns on these four are PS225, PS90, PS90 and PS36. There are two ways of looking at this. One is to say, 'What a fabulous return: our collective net wealth has grown over 10 per cent, from PS400 to PS441, so we must all be winning.' The more pessimistic viewpoint is to say, 'Sure, but most of you are now poorer than when you started, and one of you is seriously broke. In fact, the person with PS36 needs to throw three heads in a row just to recover his original stake.'

If you toss a coin three times in a row, it is true that there is only a 1/8 chance that you'll get three heads in a row. But this is confounded by the fact that you're looking at a short sequence. On average, only 14 flips are required to get three heads in a row, and in 100 flips, there's a greater than 99.9% chance there will be three heads in a row at least once.