100 Essential Things You Didn't Know You Didn't Know (18 page)

Imagine that our quest to explore the Universe has made contact with a civilisation on the strange world of Janus. The monitoring of their political and commercial activity over a long period has shown us that on the average their citizens tell the truth ¼ of the time and lie
³
/4 of the time. Despite this worrying appraisal, we
decide
to go ahead with a visit and are welcomed by the leader of their majority party who makes a grand statement about his benevolent intentions. This is followed by the leader of the opposition party getting up and saying that the Leader’s statement was a true one. What is the likelihood that the Leader’s statement was indeed a true one?

We need to know the probability that the Leader’s statement was true given that the opposition head said it was. This is equal to the probability that the Leader’s statement was true and the opposition head’s claim was true divided by the probability that the opposition head’s statement was true. Well, the first of these – the probability that they were both telling the truth is just ¼ × ¼ =
¹
/16. The probability that the opponent spoke the truth is the sum of two probabilities: the first is the probability that the Leader told the truth and his opponent did too, which is ¼ × ¼ =
¹
/16, and the probability that the Leader lied and his opponent lied too, which is
³
/4 ×
³
/4 =
⁹
/16. So, the probability
¹⁶
that the Leader’s statement was really true was just
¹
/16 ÷
¹⁰
/16, that is
¹
/10.

55

How to Win the Lottery

A lottery is a taxation – upon all the fools in creation. And heaven be praised, it is easily raised, credulity’s always in fashion.

Henry Fielding

The UK National Lottery has a simple structure. You pay £1 to select six different
numbers
from the list 1,2,3,. . ., 48,49. You win a prize if at least three of the numbers on your ticket match those on six different balls selected by a machine that is designed to make random choices from the 49 numbered balls. Once drawn, the balls are not returned to the machine. The more numbers you match, the bigger the prize you win. Match all six and you will share the jackpot with any others who also share the same six matching numbers. In addition to the six drawn balls, an extra one is drawn and called the ‘Bonus Ball’. This affects only those players who have matched five of the six numbers already drawn. If they also match the Bonus Ball then they get a larger prize than those who matched only the other five numbers.

What are your chances of picking six numbers from the 49 possibilities correctly, assuming that the machine
^fn1
picks winning numbers at random? The drawing of each ball is an independent event that has no effect on the next drawing, aside from reducing the number of balls to be chosen from. The chance of getting the first of the 6 winning numbers from the 49 is therefore just the fraction 6/49. The chance of picking the next of the remaining 5 from the 48 balls that remain is 5/48. The chance of picking the next of the remaining 4 from the 47 balls that remain is 4/47. And so on, the remaining three probabilities being 3/46, 2/45 and 1/44. So the probability that you pick them all independently and share the jackpot is

6/49 × 5/48 × 4/47 × 3/46 × 2/45 × 1/44 = 720/10068347520

If you divide this out you get the odds as 1 in 13,983,816 – that’s about one chance in 13.9 million. If you want to match 5 numbers plus the Bonus Ball, then the odds are 6 times smaller, and your chance of sharing the prize is 1 in 13,983,816/6 or 1 in 2,330,636.

Let’s take the collection of all the possible draws – all 13,983,816 of them – and ask how many of them will result in 5, or 4, or 3, or 2, or 1, or zero numbers being chosen correctly.
¹⁷
There are just 258 of them that get 5 numbers correct, but 6 of them win the Bonus Ball prize, so that leaves 252; 13,545 of them get 4 balls correct,246,820 of them that get 3 balls correct, 1,851,150 of them that get 2 balls correct, 5,775,588 of them get just 1 ball correct, and 6,096,454 of them get none of them correct. So to get the odds for you to get, say, 5 numbers correct you just divide the number of ways it can happen by the total number of possible
combinations
, i.e. 252/13,983,816, which means odds of 1 in 55,491 if you buy one lottery ticket. For matching 4 balls the odds are 1 in 1,032; for matching 3 balls they are 1 in 57. The number of the 13,983,816 outcomes that win a prize is 1 + 258 + 13,545 + 246,820 = 260,624 and so the odds of winning any prize when you buy a single ticket are 1 in 13,983,816/260,624, that is about 1 in 54. Buy a ticket a week with an extra one on your birthday and at Christmas and you have an evens chance of winning something.

This arithmetic is not very encouraging. Statistician John Haigh points out that the average person is more likely to drop dead within one hour of purchasing a ticket than to win the jackpot. Although it is true that if you don’t buy a ticket you will certainly not win, what if you buy lots of tickets?

The only way to be sure of winning a lottery is to buy
all
the tickets. There have been several attempts to use such a strategy in different lotteries around the world. If no jackpot is won in the draw, then usually the unwon prize is rolled over to the following week to create a super-jackpot. In such situations it might be attractive to try to buy almost all the tickets. This is quite legal! The Virginia State Lottery in the USA is like the UK Lottery except the six winning numbers are chosen from only 44 balls, so there are 7,059,052 possible outcomes. When the jackpot had rolled over to $27 million, Australian gambler Peter Mandral set in operation a well-oiled ticket buying and printing operation that managed to buy 90% of the tickets (a failure by some of his team was responsible for the worrying gap of 10%). He won the rollover jackpot and went home with a healthy profit on his $10 million outlay on tickets and payments to his ticket-buying ‘workers’.

^fn1
Strictly speaking there are 12 machines (which each have names) and 8 sets of balls that can be used for the public draw of the winning numbers on television. The machine and the set of balls to be used at any draw are chosen at random from these candidates. This point is usually missed by those who carry out statistical analyses of the results of the draw since the Lottery began. Since the most likely source of a non-random element favouring a particular group of numbers over others would be associated with features of a particular machine or ball, it is important to do statistical studies for each machine and set of balls separately. Such biases would be evened out by averaging the results over all the sets of balls and machines.

56

A Truly Weird Football Match

Own goal: Own goals
tend, like deflections, to be described with sympathy for those who fall victim to them. Often therefore preceded by the adjectives
freak
or
bizarre
even when ‘incompetent’ or ‘stupid’ might come more readily to mind.

John Leigh and David Woodhouse,
The Football Lexicon

What is the most bizarre football match ever played? In that competition I think there is only one winner. It has to be the infamous encounter between Grenada and Barbados in the 1994 Shell Caribbean Cup. This tournament had a group stage before the final knockout matches. In the last of the group stage games Barbados needed to beat Grenada by at least two clear goals in order to qualify for the next stage. If they failed to do that, Grenada would qualify instead. This sounds very straightforward. What could possibly go wrong?

Alas, the law of unforeseen consequences struck with a vengeance. The tournament organisers had introduced a new rule in order to give a fairer goal difference advantage to teams that won in extra time by scoring a ‘golden goal’. Since the golden goal ended the match, you could never win by more than one goal in such a circumstance, which seems unfair. The organisers therefore decided that a golden goal would count as two goals. But look what happened.

Barbados soon took a 2–0 lead and looked to be coasting through to the next phase. Just seven minutes from full time Grenada pulled a goal back to make it 2–1. Barbados could still qualify by scoring a third goal, but that wasn’t so easy with only a few minutes left. Better to attack their own goal and score an equaliser for Grenada because they then had the chance to win by a golden goal in extra time, which would count as two goals and so Barbados would qualify at Grenada’s expense. Barbados obliged by putting the ball into their own net to make it 2–2 with three minutes left. Grenada realised that if they could score another goal (at either end!) they would go through, so they attacked their own goal to get that losing goal that would send them through on goal difference. But Barbados resolutely defended the Grenada goal to stop them scoring and sent the match into extra time. In extra time the Barbadians took their opponents by surprise by attacking the Grenada goal and scored the winning golden goal in the first five minutes. If you don’t believe me, watch it on YouTube!
^fn1

^fn1
http://www.youtube.com/watch?v=ThpYsN-4p7w

57

An Arch Problem

Genius is four parts perspiration and one part having a focused strategic overview.

Armando Iannucci

An old arch of stones can seem a very puzzling creation. Each stone looks as if it has been put in place individually, but the whole structure looks as if it cannot be supported until the last capstone is put in place: you can’t have an ‘almost’ arch. So, how could it have been made?

The problem is an interesting one because it is reminiscent of a curious argument that is much in evidence in the United States under the name of ‘Intelligent Design’. Roughly speaking, its advocates pick on some complicated things that exist in the natural world and argue that they must have been ‘designed’ in that form rather than have evolved by a step-by-step process from simpler forms because there is no previous step from which they could have developed. This is a little subjective, of course – we may not be very imaginative in seeing what the previous step was – but at root the problem is just like our arch, which is a complicated construct that doesn’t seem to be one step away from a slightly simpler version of an arch with one stone missing.

Our unimaginative thinking in the case of the arch is that we have got trapped into thinking that all structures are built up by adding bits to them. But some structures can be built by subtraction. Suppose
we
started with a heap of stones and gradually shuffled them and removed stones from the centre of the pile until we left an arch behind. Seen in this way we can understand what the ‘almost’ arch looks like. It has part of the central hole filled in. Real sea arches are made by the gradual erosion of the hole until only the outer arch remains. Likewise, not all complexity in Nature is made by addition.

58

Counting in Eights

The Eightfold Path: Right view, right intention, right speech, right action, right livelihood, right effort, right mindfulness and right concentration.

The Noble Eightfold Way

We count in ‘tens’. Ten ones make ten, ten tens make a hundred, ten hundreds make a thousand and so on forever. This is why our counting system is called a ‘decimal’ system. There is no limit to its scope if you have enough labels to name the results. We have words like million, billion and trillion for some of the large numbers, but not for every one that you might need to write down. Instead we have a handy notation that writes 10
ⁿ
to denote the number which is 1 followed by n noughts, so a thousand is 10
³
.

The origin of all these tens in the counting system is not hard to find. It is at our fingertips. Most ancient human cultures used their fingers in some way for counting. As a result you find counting systems based on groups of five (fingers of one hand), ten (fingers of both hands), groups of twenty (fingers plus toes), or mixtures of all or some of these systems. Our own counting system betrays a complicated history in which different counting systems merged to form new ones by the presence of old words that reflect the previous base. Thus we have a word like ‘dozen’, for 12, or ‘score’ (derived from the old Saxon word
sceran
, meaning to shear or to
cut
) for 20, with its interesting triple meaning of 20, to make a mark or to keep count. All three meanings reflect the time when tallies were kept on pieces of wood by marking (scoring) them in groups of 20.

Despite the ubiquity of the base 10 counting system in early culture, there is one unusual case where a Central American Indian society used a base 8 counting system. Can you think why this might be? I used to ask mathematicians if they could think of a good reason, and they usually responded by saying that 8 was a good number, to use because it has lots of factors, it divides exactly by 2 and 4, so you can divide portions into quarters without creating a new type of quantity that we call a fraction. The only time I got the right answer though was when I asked a large group of 8-yearold children and one girl immediately produced the answer: they were counting the gaps between their fingers. If you hold things between your fingers, strings or pieces of material, this is a natural way to count. The base eighters were finger counters too.