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Lesson 2

Betting on horses and football is a huge business in Las Vegas.  While off-track betting is illegal in rest of the United States, a good deal of money finds its way to Nevada.  I must admit that the set-up in the Nevada sports books, as betting shops are called, is the nearest thing to gambling heaven yet devised.

            Imagine a lecture hall, with long rows of seats, each with its own table top, banked up around a wide semi-circle.  Down below, across the front of the room, is a line of six or eight television monitors showing, thanks to satellite, every race meeting, every football game, every important sporting event, taking place in the United States.  To the side of the TV screens are the betting windows.
            Imagine, further, that while you sit there, studying the form and racing sheets and all the other paraphernalia available, that you order up any drinks you like, on the house.  What a deal! It ’s like being a student all over again but without any grades.  Only one tiny snag … you glottal pick the winner.
Imagine, further, that while you sit there, studying the form and racing sheets and all the other paraphernalia available, that you offer up any drinks you like, on the house. What a deal! it's like being a student all over again but without any grades. Only one tiny snag; you glottal pick the winner.
            You want a line on a football match, a baseball game, college basketball or any other event coast to coast, the sports book has got it.  As one of the leading odds makers in town, Michael Roxborough, known everywhere as Roxy, explained it to me, the line is calculated on the basis that each bettor wages 11 to win 10, which gives the bookmakers a theoretical edge of 4 ½ per cent.  (The bookmaker takes in 22 dollars and pays out 21, leaving a profit of 1/22 = 4.55 per cent.)
            The line reflects the odds maker’s assessment of the teams, but is also cunningly designed to look attractive to supporters of both sides, to encourage betting on each.  The sports books work to a fine margin, but they do have an ace in the hole: the set-up in the casinos is located close to the gaming poker one table.  Anyone clever or lucky enough o call it right and walk away with a fistful of dollars has got to navigate his way past the tables on the way out.
            Sports betters are not the only ones given to self-delusion.  Another aspect of anchoring is that people tend to over-estimate the probability of events which are jointed together (conjunctive events)  and under-estimate events which seem to be separate (disjunctive events).
  This kind of bias is especially prevalent in political life.  It leads to over-optimism in assessing the likelihood that a plan with many steps in it (like managing the economy) will succeed or a long-range project be completed on time.  Conversely it leads to under-estimation of the risk that, say, a nuclear reactor, dependent on many separate components, will malfunction (look at Chernobyl or the US space programmer) or for that matter, that a complex system like one’s own body will continue in good working order.

            Conjunctive events make good sucker bets.  One of the most familiar, which I repeat here in case you don’t know it, concerns how many people need to be in the same room before it is better than an even money bet that at least two of them share the same birthday.  The answer, which confounds most people’s intuitive reaction, based on the fact that there are 365 days in the year, is surprising 23.
            The probability of a second person in the casino room having the same birthday as the first person is 1/365, so the probability of having a different birthday is 364/365.  The probability for the third person having a different birthday from each of the other two is 363/365, for the fourth person 362/365, and so on.  The probability that they will all have different birthdays is the product of all these individual probabilities, i.e. 364/365 x 363/365 x 362/365 x …., and by the time there are 23 people in the room, this sum works out at just under 0.5.  So, with 23 people in the room it is slightly more than 50 per cent likely that at least two of them will have the same birthday.
            The  essential point which intuitive judgment misses in assessing conjunctive events is that when a lot of probabilities are multiplied together, even if they are all very near to 1 (i.e. the likelihood of each individual case is close to 100 per cent), their product gets smaller very quickly.  By contrast, in disjunctive events intuition tends to underestimate probabilities: if every single component in an aeroplane is 99.9999 per cent reliable, and there are half a million components in the plane, the poker probabilities of at least one component breaking down is approximately 500,000 x 0.0001 per cent which is 50 per cent.
            The argument put forward by psychologists like Testy and Kahn man (Whose findings I have considerably over-simplified) is that an analysis of the heuristics that a person uses in judging the probability of an event may reveal whether his judgment is likely to be too high or too low.  Subjective probabilities play an important role in everyone’s life.

            The decisions we make, the conclusions we reach, and the explanations we offer are usually based on the judgments of the likelihood of uncertain events such as success in a new job, the outcome of an election, or the state of the market … Although no systematic theory about the psychology of uncertainty has emerged… perhaps he most general conclusion obtained from numerous investigations is that people do not follow the principles of probability theory in judging the likelihood of uncertain events.

            Biases that people have are inherent in their way of judging things, not a product of differences in motivation, such as wishful thinking or monetary reward.  Experienced people, trained in statistics, may avoid elementary errors such as the casino gamblers fallacy, but they also go wrong when they think intuitively about more complex problems.
            The reason why we do not learn what our personal biases are from everyday experience is that the relevant information isn’t in a form that can be easily checked out.  If one kept a record of events that actually occurred, measured against the probability that one assigned to them in advance, that would help.  But that is not the way people think.
  If it were one might discover, for example, that only 50 per cent of the predictions to which one gave a very high probability, say 0.9, came true.  (In my own work, as a journalist commenting on foreign affairs, I have found that one tends to savor the times that one calls it right like making a winning bet but tends even more to have total amnesia over the many other times one gets it wrong.  The prudent newspaper reader or TV viewer makes, instinctively, an estimate of the reliability, that is the probability, of the journalist ’s or broadcaster’s accuracy.  When it comes to official Government statements, people frequently assume the opposite of what is claimed is the truth!)

            Yes, to err is human, and to err in assessment to probability is all too human.  That is what the human mind is like, an uncertain instrument.  Of course one may say that, given a little practice, given a few lessons in probability poker theory, one wouldn’t make these mistakes.  Not so: experts make mistakes too.  Besides, gamblers do not normally think in this way.  Quite the reverse: they operate on a hunch, feel, lucky numbers, a sense of the dice being hot, intuitive feelings, omens, random happenings about them, flying high on a wing and a prayer. ‘Although a rational mind (as playwright Jack Richardson has put it) must insist that the grimace of a stranger or a moment of intestinal agony has no causal link to the way cards will fall, it is nevertheless true that a gambler’s instinct comprehends relations between events that are perhaps too subtle for ordinary modes of observation.  It is this instinct on which his survival is based, for if he ignores it … he becomes … someone doomed to be unloved by fortune and destroyed by mathematics.’  (Memoir of a Gambler, 1979.)
True gamblers go further, deliberately ignoring probabilities in favor of their gut feelings.  Is there a man so base who will not forever cherish the memory of the great Sky Master son, nonchalant hero of Damon Run yon;s story The Idyll of Miss Sarah Brown?  Of all the high players this country ever sees, there is no doubt but that the guy they call The Sky is because he goes so high when it comes to betting on any proposition whatever.  He will bet all he has, and nobody can bet any more than this … and furthermore The Sky never wishes to play with any the best of it himself, or anyway not much … The Sky is strictly a player… as far as The Sky is concerned, money is nothing but just something for him to play poker with and the dollars may as well be doughnuts as far as value goes with him.’

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