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ONE  of the most intriguing aspects of gambling guide is how people make up their minds when they are taking chances.  At its most basic, how do you decide whether to back red or black?  Why do you choose one number over another?  How do you assess the various factors in more complicated decisions in taking chances?  How good is people’s judgment in uncertainty?

            According to academic studies on these matters, many decisions are based on beliefs concerning the likelihood of uncertain events such as the outcome of an election, the guilt of a defendant, or the future value of the dollar: beliefs usually expressed in statements such as:  ‘I think that … ’ ‘Chances are … ’, ‘It is unlikely that … ’, and so forth.  Occasionally beliefs concerning uncertain events are expressed in numerical form as odds or subjective probabilities.
            People rely on a limited number of heuristic (problem solving) principles, which reduce the complex task of assessing probabilities and predicting values to simpler operations of judgment.  In general these methods are quite useful, but sometimes they lead to severe and systematic errors.  (Judgment under Uncertainty:  Heuristics and Biases, amos Testy and Daniel Kahn man, Science, 1974).
One of the most common mistakes is known as the gamblers fallacy’: the belief that the law of large numbers applies also to small numbers.  People expect that a sequence of events in a random process – say the tossing of a coin – will represent the essential characteristics of the process even when the sequence is short.  In tossing a coin heads or tails people regard the sequence  H-T-H-T-T-H as more likely than the sequence H-H-H-T-T-T, because the latter does not appear to be random, and more likely than the sequence H-H-H-H-T-H which does not seem to reflect the ‘fairness’ of the coin.
  Thus people expect the essential characteristics of the process of a very long series of tosses to be shown in each of its parts.  Look at the roulette players in any casino in the world! (I am ashamed to confess that the first time I went to Las Vegas, when I was old enough to know better, I had a system based on waiting for a sequence of eight reds before backing black; it took less than two hours for the system to be shredded.)  In other words, people view chance as a self-correcting process in which a deviation in one direction induces a deviation in the opposite direction to restore the balance.  In fact deviations are not ‘corrected’ as a chance process goes on, they are merely diluted.

            All right, but this habit of mind is for simple folk, not one which sophisticated people adopt.  Not so, according to Testy and Kahn man, who found that poker experience research psychologists had a lingering belief in the ‘law of small number’ too.  In this particular  heuristic or method of judgment, known as ‘representative ness’, an event is judged probable to the extent that it represents the essential features of the structure from which it originates.  Probability is judged by similarity.
            Very many questions of probability belong to the type: What is the probability that event A belongs to class B?  Or what is the probability that process B will generate event A?  In answering such questions people typically estimate the probability by the degree to which A is representative of B.
            For instance, suppose you are trying to predict the future value of a stock or a commodity.  Given a description of a company which is very favorable, a very high profit will appear most representative of that description; if the description is mediocre, a mediocre performance will appear most representative.  What people overlook is the degree to which it permits accurate prediction.  So if people make investments, or bets, solely on the basis of the favorableness of the description, their decision will not take into account the reliability of the evidence and the expected accuracy of the prediction. This mode of judgment violates statistical theory: if the descriptions of companies provides no information relevant to profit, then average profit should be predicted.
            In another heuristic known as ‘Availability’, probability is judged by association, or what comes easily to mind.  For example, the divorce rate in a given community may be assessed by recalling divorces among one’s acquaintances. The sight of a card games overturned on the side of the road certainly gives one a heightened sense of the risks of driving.  In other words the estimate of probability is influenced by the way in which instances or associations seem to run in parallel.  A doctor who has heard a patient complain that he is tired of life and wonders whether that patient is likely to commit suicide may recall similar patients he has known.  What the doctor ought to think about is patients who resemble the present case and attempt suicide: the relevant statistic is the frequency of attempted suicides in this class.  The example doesn’t have to be a doctor.  ‘We know of no reason to believe that the intuitive predictions of stockbrokers, sportscasters, political analysts or research psychologists are less susceptible to biases, ’ say authors.
            Essentially, what is in question is the gap between subjective thinking and cognitive thinking.  Subjective thinking resembles assessment of physical phenomena like distance or size.  For instance, the apparent distance of an object is determined by its clarity: the more sharply it is seen, the closer it appears to be.  However, reliance on this rule leads to systematic errors.  Distances are often over-estimated when visibility is poor because the contours of objects are blurred; on the other hand distances are often under-estimated when visibility is good, because objects are seen sharply.  Such biases are also found in the intuitive judgment of poker probabilities.
            Here’s example which you can test for yourself.  In many situations people make estimates by starting from an initial value that is adjusted to yield the final answer, a method of judgment termed ‘adjustment ’ or ‘anchoring’.  Typically, adjustments are insufficient at both ends of the scale.  And different starting points yield different estimates, which are biased towards the initial value.  Take five seconds to estimate the total product of this line of numbers going from left to right:

8  x 7 x  6  x  5  x  4  x  3 x  2 x 1

then five seconds to estimate

1 x  2 x  3 x  4  x 5 x  6 x  7 x  8

or try it on any two acquaintances with ten seconds to spare.
            To answer such questions rapidly, people perform a few steps of computation and estimate the final total by extrapolation or adjustment.  Because the results of the first few steps of multiplication are higher in the descending sequence than in the ascending one, the former is judged higher than the latter.  When the test was given to high school students, the median answer for the ascending sequence was 512, while the median answer for the descending sequence was 2,250.  What did you get?  The correct answer is 40,320.
            It is also worth nothing, so far as gambling is concerned, that different people have different feelings about the same bet.  In other words that there is a subjective estimate of value.  Two quarters risked on a pinball machine by a pair of teenage boys may be every bit as significant to them as twenty-five dollars doubled down on a blackjack hand to their parents on a night out at a casino, while the same twenty-five chip might be flipped to the cocktail waitress by a gambler betting in units of five hundred.  It ’s all according.  A still further distinction exists between this subjective value of the bet itself and the subjective value or ‘pleasure’ afforded by the actual process of playing the machine or playing the blackjack hand – positive or negative – in the excitement and anxiety it arouses.  As has been pointed out, we all have different base-lines from which to measure risk and opportunity.  So quite apart from the odds, there are three elements in a bet: its real value, its subjective worth and its excitement quotient.
            To take a personal illustration: I am a member of a group of friends who play five-minute chess for 50 pence a game.  The monetary result is insignificant, the price of a cup of coffee; but the subjective value of winning those games goes beyond mere price, and I have witnessed more rows at the chessboard over these trivial sums than I ever have at the poker table playing for hundreds.
            When the big money players in Las Vegas play a game of golf, for instance, the amounts wagered are by any ordinary standards, hysterical.  A $ 10,000 Nassau – ten grand on the first nine holes, ten on the second, and ten on the overall result – is merely for starters.  With side bets, a round can easily get up to six figures.  The players simply wouldn’t ‘feel’ it playing for hundred.
            I have been particularly struck by the colossal bets which the top poker players like to wager on football and other sports, because, properly speaking, these men are not ‘gamblers’ in their chosen calling.  They play a game of skill and depend on their talent at it to survive, like chess masters.  Anyone among them can reel off percentages down to three places of decimal, on the probabilities of a survive, like chess masters.  Anyone among them can reel off percentages down to three places of decimal, on the probabilities of a particular poker hand of cards.  Yet in sports bets they go wild.  They wager $ 50,000 or $ 100,000 on events which are not only inherently unpredictable, and on which the odds are against them, but on which their own evidence (going back to the ‘representative’ heuristic) is hopelessly subjective.  What happens is that one team ‘looks good’, while the rival quarter-back couldn’t throw a tea-towel.  Or the full back had a problem with his girlfriend last week, but now they’re back together.  ‘I tell ya, I wouldn’t give a wooden nickel for that bunch of no-hopers from the East Coast, ’ and so on.  Evidence?  Schmevidence.

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