I was musing on the Author's thesis that on occasions commonsense experience conflicts with logical deduction, said the Statistician,
when I recalled an example from my own field of study - a calculation of probability that all statisticians would agree on, but none would act upon. If the Author will allow me, I shall relate it as the tale of:
There I am, walking down the street wondering whether to spend a bob or two in Sally Lunn's Bijou Caff and Grill on the offchance that congenial company may be found therein, or whether to invest it in a nag called Wingless Wonder that my milkman informs me that very morning is a sure thing for the 2.30, on the solemn oath of his brother-in-law who is acquainted with a girl whose sister's boyfriend cleans stables in Epsom, when whom do I see approaching but System Sid, wearing a big cigar between his teeth, a broad smile on his mug, and a buxom blonde on his arm. I am very surprised indeed to see Sidney in this condition, because when we last meet he is stony broke and most downhearted, so I greet him cordially but not so loudly as to be audible to any passing fuzz, as I figure that sudden affluence is often a sign of characters whose assistance is required by the Old Bill in connection with their enquiries.
Things all right then, are they, System? I ask, sotto voce, to which he replies in a hearty manner,
Never better, squire, never better. Perhaps you are wondering about my change of circumstances since our last encounter? You will recall that on that occasion I am still licking my wounds after my grand challenge match with Educated Ernie? I nod my head, for I do indeed recollect just that, the match in question being regarded in these parts as most noteworthy, the circumstances being as follows.
System Sid is well known as a regular punter who bets on any game of chance, and indeed upon many in which chance is not allowed to interfere. He is always following some system or other, which is how he acquires his monicker, but with infrequent success, perhaps because his idea of a system is apt to verge upon the eccentric. Last year, for example, he loses an unwarrantable proportion of his dole money backing second favourites who are wearing blinkers for the first time.
Sidney has dreams of a big win someday, but his chief ambition is to put one over on Educated Ernie, a character who is so called because when he is last a guest of Her Majesty's Government he enrols in the Open University, thereby acquiring both a higher education and maximum remission. He puts his name down for Sociology like all the other cons, but a clerical error in the Governor's office results in his taking a degree in Statistics, which is close enough, alphabetically speaking. To his surprise, he finds that a large part of Statistics concerns probability theory, and that a large part of probability theory concerns the outcome of games of chance. Upon his release, Ernie puts his new-won knowledge to good use, and soon has a reputation for knowing the odds against drawing to a bobtail straight, and suchlike useful intelligence. His studies obviously include Applied Statistics, because in next to no time Ernie is winning more bets than Sidney loses, which is saying something.
The rivalry between them reaches a head when Sidney challenges Ernie to a series of personal wagers, which Ernie gladly accepts. This is the match to which Sidney now refers, and he reminds me of its details.
We agree, he says,
to play three different games, each game to be played continuously for a whole day. The first day, Ernie suggests a simple dice game. I stake £6 and throw three dice. If any of them shows a 5, Ernie pays me £13. I figure that I have a 50-50 chance of getting a 5, so the even money payout would be £12. With the extra pound as my edge, how could I lose? To my surprise, at the end of the first day, I am out of pocket.
The second day, Ernie bets me even money that in any group of 30 strangers, two or more of them will have the same birthday. I figure that as there are 365 days in a year, the odds are more than 10 to 1 against, so happily accept this bet. To my sorrow, I find I am paying out to Ernie far more often than I am winning.
The third day, Ernie suggests that we sit in the window of Sally Lunn's and watch the traffic go by. He bets me even money that whenever a blue van goes past, the next bus will be a number 57. Naturally, I check with the bus company before I take this bet. They tell me that there are three bus routes passing Sally Lunn's: the 57, the 23, and the 79. The frequency of all three is exactly the same: one every five minutes throughout the day. I also observe that blue vans are by no means uncommon in that street; one passes every few minutes. Naturally I figure that the odds are 2-1 against the next bus being a 57, and I take the bet. Would you believe that after 5 hours, Ernie has won 35 times and I have won 25 times and am out of cash? Ernie is by no means magnanimous in victory. He not only refuses to lend me the bus fare home, but he calls me a mug for accepting such bets. He claims that he figures the odds for each game and that they favour him. Furthermore, he says, he looks forward to meeting more mugs like me, for he is always willing to wager on bets where the expected value of a win exceeds his stake.
Naturally, I am very disheartened by this debacle, says Sidney,
but a few days later my luck changes. He goes on to tell me that he pops into the betting shop to pass the time of day with his coterie, and there meets a peculiar little man who projects such a receptive aura that soon Sidney is confiding his unhappy experience to him, whereupon the stranger clucks sympathetically.
I fear that this Ernest out-calculates you, he says.
In the first game, the fair payout is nearer £14 than £13. The odds are not 50-50, as you suppose. You are forgetting that sometimes you are getting more than one 5 at a time, and for these you get only one payout. The probability of not getting a 5 with three dice is (5/6)3, or approximately 4 out of 7. That was Ernie's edge.
The birthday bet is somewhat similar. The probability of 30 people having birthdays all on different days is:
(364/365) x (363/365) x (362/365) x .......................x (336/365) ,
or a little less than 30%, so you see Ernie is on to a good thing with an even money bet.
As to the buses, he says,
you are overlooking that what counts is not how often they run, but when, in relation to each other. As a matter of fact, he continues,
I happen to know that the 57 runs on the hour (and every 5 minutes thereafter), while the 23 runs one minute later, and the 79 one minute after that. So you see, he says,
in any five minute period, there are three minutes when the next bus will be a 57, and only two minutes when it will be a 23 or a 79.
Sidney is very sad to receive this information, as he begins to appreciate that he is indeed a mug, and he says despondently to the stranger,
So there is no way of betting successfully against a person who figures the percentages as accurately as Educated Ernie?
I am not saying that, says the stranger pensively.
You tell me, I recall, that Ernest promises faithfully to bet on any wager where the odds are in his favour? Then listen carefully. And he gives Sidney some very detailed instructions.
The next day, Sidney tells me, he calls in some favours to raise a little betting money, and seeks out Educated Ernie. He reminds Ernie of his promise to take any bet with favourable odds, tells him that he now has such a proposition for him, and asks him to make good.
Certainly, says Ernie,
on condition that you prove that the odds are favourable, and that I am allowed to play the game as often as I wish. Sidney agrees, and then speaks as follows.
First, says Sidney,
let us be sure we agree on how to calculate your expectation of a game. For example, if you are to throw a single dice, and I am to pay you one pound per spot showing, you have a one in six chance to win £1, plus a one in six chance to win £2, plus a one in six chance to win £3, and so on. So your expectation is
Ernie assents to this method of calculation.
(1/6x£1) + (1/6x£2) + (1/6x£3) + (1/6x£4) + (1/6x£5) + (1/6x£6)
which is £3.50. So the odds are in your favour if I offer to play such a game with you for a stake of less than £3.50.
Very well, says Sidney,
here is the game I propose we play: you pay a stake and then toss a coin until it comes down heads, which is the end of the game. If you get a head on the first throw, I pay you £2; if you do not get a head until the second throw, I pay you £4; if you do not get a head until the third throw, I pay you £8; and so on, doubling the payout for every successive tail that you throw. Now what is your expectation for this game? Obviously you have a chance of one in two of winning £2, plus a chance of one in four of winning £4, plus a chance of one in eight of winning £8, and so on, so your expectation is:
(1/2x£2) + (1/4x£4) + (1/8x£8) + (1/16x£16) + .......... and so on ad infinitum,
which equals £1 + £1 + £1 + £1 + ..........and so on ad infinitum. Since your expectation is an infinite sum of money, it follows that no matter how large your stake, the odds are in your favour. I am willing to let you play this game as often as you like, for a stake of £10,000 per game, which is infinitely less than your expectation. When do you want to begin?
So you win a large sum of money from Ernie playing this game? I ask, hoping to expedite Sidney's narrative.
Not exactly, says Sidney.
At first Ernie goes a strange green colour as he checks the calculations and can find nothing wrong with them. Then he offers me £5,000 to release him from his promise to wager with me, which I accept, as I am by no means as small minded in victory as he is. Then he asks me where I learn of this game, and I tell him about the stranger. He asks his name, and I remember that the stranger tells me that his name is Blaze Paskle. Educated Ernie goes from green to white when he hears that name, for it appears that not only is this Paskle fellow a Frenchman very well known for being adept at calculating probabilities in wagers, he is also dead for the past three hundred years.
To tell you the truth, adds Sidney,
I am wishing that I know earlier that Blaze is not of this world, for then I might have more confidence in him and save myself the fifty quid that I bung Fingers McShane for the loan of his Irish penny, which he positively guarantees to turn up heads three times out of four.
When the tale ended, we turned to the Mathematician for confirmation of the calculations involved. "There is no error in the computation of an infinite expectation for the game, which was invented by Nicolaus Bernoulli in the 18th century, not by Blaise Pascal in the 17th." (He glared at the Statistician, who muttered something about poetic licence.) "The paradox arises because of our failure to appreciate the gulf that lies between the theoretical and the practicable. It is akin to one of the oldest gambling fallacies, namely that you can guarantee to win on the horses simply by increasing your stake so as to cover any previous losses. Theoretically it is true; sooner or later you will come out ahead, but the snag is in that later. In practice, you run out of money first. The coin tossing game suffers from the same limitation. The infinite expectation includes possible outcomes whereby the winner receives more money than there is in the world and which may not occur in a thousand years of continuous play. In practice, the game would necessarily be subject to constraints of time and money."
"Precisely," the Statistician agreed. "If a casino offered this game to its customers, it would need to set a maximum payout for any one game. Suppose the limit were as low as $16. (Let us assume that the game is now defined in dollars rather than pounds, out of deference to Las Vegas.) The player's expectation would then be:
(1/2x$2) + (1/4x$4) + (1/8x$8) + (1/8x$16),giving a fair stake of $5. If the limit were increased to $32, the expectation would be:
(1/2x$2) + (1/4x$4) + (1/8x$8) + (1/16x$16) + (1/16x$32),giving a fair stake of $6. For each doubling of the limit, the fair stake increases by $1. The relationship between the limit, L, and the fair stake, F, is:
"Do you remember the old story of the man who invented chess, and gave the game to the Emperor of China? The Emperor allowed him to name his own reward, and he asked for a single grain of rice for the first square on the board, two grains for the second, four grains for the third, and so on, doubling the number of grains for each of the 64 squares. The Emperor protested that this was inadequate, but on the man's insistence, ordered the Imperial warehouse to provide the rice. They found to their amazement that this simple order would require more than all the rice in the world to fulfil a thousandth part of it. The story illustrates the fierceness of the exponential progression involved in doubling up, and demonstrates the nonsense of referring to an infinite stake. For a stake of $31 in the St Petersburg game, the fair limit would be more than $1bn. For a stake of $41 the fair limit would exceed the Gross National Product of the United Kingdom. Increase the stake to $44, and the fair limit would exceed the GNP of the United States by some 60%. Long before the stake reached three figures a fair limit would exceed all the money in the world."
After a while, the Philosopher spoke. "The thought that something whose worth in theory is beyond price can, by the application of the constraints of the real world, be reduced in value to less than the balance of my meagre bank account, is strangely disturbing. Either I am worth more, or the world is worth less, than I have hitherto supposed." His words appeared to have a sobering effect upon the company, which dispersed, each reflecting on the state of his finances, and one or two, perhaps, wondering where they might find an Educated Ernie.