By Michael Carroll
Best-selling author of All for One, One for Sorrow, Two for a Quid, Three for Five Six

Introduction     Part 1     Part 2     Part 3     Part 4     Part 5

Lesson 6: Understanding Numbers

Now, you may all think that you understand how numbers work, but you're wrong. Probably. Unless, of course, you've already read this article. And understood it.

Two out of Three

This is one of those strange ones: say you've got a coin and you're using it to help you make a decision: "If it's heads, I'll have another slice of pizza. If it's tails, I won't." So you flip the coin and it comes up tails. "Rats!" you think, because you really do want that slice of pizza, and you were hoping for the coin to come up heads because you want to justify your gluttony by being able to blame it on the Fates. So, you decide "Best two out of three," and you flip the coin again, hoping for two heads.

The incorrect assumption here is that "best two out of three" gives you a better chance of getting the answer you really want than just one coin-flip... But let's look at the facts:

Assuming for the moment that the chance of any coin coming up heads or tails is fifty-fifty on any given flip, then there are eight possible outcomes for three coin flips:

1. Heads, heads, heads
2. Heads, heads, tails
3. Heads, tails, heads
4. Heads, tails, tails
5. Tails, heads, heads
6. Tails, heads, tails
7. Tails, tails, heads
8. Tails, tails, tails

Since 2, 3, and 5 are the same (two heads, one tails), and 4, 6 and 7 are the same (one heads, two tails), we can narrow this down to four possible outcomes (because we don't care in which order the heads or tails appear):

1. Heads, heads, heads
2. Heads, heads, tails
3. Heads, tails, tails
4. Tails, tails, tails

Now, in our pizza-eating scenario, we're hoping for outcome 1 or 2 (more heads than tails). Therefore, the odds of the Fates allowing us to have that extra pizza slice are the same as the odds of the Fates deciding that we need to watch our weight.

Some people have a tendency to keep going until the coins land the way they want. If you're one of those people, you might as well just not bother with the coin-tossing and eat the pizza. Remember: if the Fates do exist, then they're not going to take kindly to you blaming them for any weight you might gain.

Conclusion: The "Best Two out of Three" approach is only useful if you decide to employ it after the first flip fails to give you the answer for which you are looking.

Majorities

We all know what a majority is, right? Wrong! Most of us don't understand it! Most of us think that a majority means "more than half" but it can't be: that wouldn't work in, say, political elections, would it? No, a majority is the "major part". I.e., more than any other part.

Say you have ten cars (maybe you're rich, or a car salesman, or something. A rather successful joy-rider, perhaps). You have four red cars, three blue cars and three green cars. The majority of your cars are red, because four is larger than three. However, since there are more non-red cars (i.e., six) than red cars, it's also true to say that the majority of your cars are not red.

So we have two contradictory statements that are both true. How can that be? Well, as we'll see, numbers aren't exactly fair...

Evens and Odds

Right, you all know what an even number is: it is a number that is divisible by two without any bits left over. 0, 2, 4, 6, 8, etc., all the way to infinity. Odd numbers are those that are not easily divisible by two: 1, 3, 5, 7, 9, and so on.

You would think that there are an equal amount of odd numbers and even numbers, but check this out:

1. If you add two even numbers, the answer is an even number.

2. If you add two odd numbers, the answer is an even number.

3. If you add an even number and an odd number, the answer is an odd number.

Since there is an infinite number of numbers, logically there is an infinite number of even numbers and an infinite number of odd numbers. But the above statements suggest that any two randomly-picked numbers will be more likely to add up to an even number than an odd number! This means, of course, that there are more even numbers than odd numbers.

"Ah," you say, "I see the flaw in that argument! You're forgetting that statements 1 and 2 have only half the number of numbers to play with that statement 3 has!"

Yeah, right. Half of infinity is still infinity, pal, so I'm still right! Since we don't have the space to examine every single number, let's do a controlled test on the first six numbers, 0 to 5.

1. Two even numbers give us six possible combinations:
0+0=0
0+2=2
0+4=4
2+2=4
2+4=6
4+4=8

2. Two odd numbers also give us six combinations:
1+1=2
1+3=4
1+5=6
3+3=6
3+5=8
5+5=10

3. However, in that range, an even and an odd number give us only nine combinations:
0+1=1
0+3=3
0+5=5
1+2=3
1+4=5
2+3=5
2+5=7
3+4=7
4+5=9

Since we know that six combinations plus six combinations is greater than nine combinations, the only conclusion to which we can come is that there are more even numbers than odd numbers.

And if you think that this result is just some weird property of addition, consider multiplication: two even numbers multiplied become an even number. An even and an odd number multiplied also become an even number. The only way to get odd numbers through multiplication is through two odd numbers. So, again, there are more even numbers than odd numbers.

Rounding

Ah, rounding numbers! What fun we all have when we go shopping and discover a CD we wants that's only £9.99! Somehow, £9.99 seems significantly less than £10.00, even though we know that they're practically the same figure. I mean, with that measly one penny left over, we'd have to buy an awful lot of CDs for the perceived price to actually make any difference to our bank accounts.

So if 9.99 is really 10.00, then this can be applied to other situations: if your birthday is, say, on November 30th, then really your birthday is in December (thanks to my six-year-old friend Isobel Murray for discovering that one!).

Rounding numbers also gives us this situation: When rounding, it's traditional that if the number ends with a five or greater, the number is rounded up, otherwise it's rounded down...

7.8 rounds to 8
7.2 rounds to 7

Okay, that's all very well, but how about this number: 4.444445. Well, that rounds to 4.44445. But since the new last digit is now a five, then that means it should really be rounded to 4.4445. And again: 4.445. And again: 4.45. This becomes 4.5, which, obviously, rounds up to 5. Now, if we don't stop there, it's pretty clear that the number becomes 10.

Likewise, 4.444444 becomes 4.44444, then 4.4444, then 4.444, 4.44, 4.4, then 4, and finally it rounds to 0.

Therefore,
4.444445 = 10
4.444444 = 0

Random Numbers

Can some numbers be more "lucky" than others? This is a question that has vexed numerologists - and Lotto players - since the noon of time. There's a lot of superstition about numbers, especially among gamblers. This is mostly because human beings are in general very, very bad at pattern recognition, but very, very good at not understanding that they're bad at pattern recognition.

For example: If you flip a coin ten times, and each time it comes up heads, what are the chances that on the next flip it'll come up tails? Well, this is how the average person might see it: "Since there's an equal possibility that on any given flip the coin will be either heads or tails, then logically the eleventh flip is more likely to be tails than heads, because that way it'll all sort of balance out a bit."

So let's experiment: I'm going to flip a coin ten times, and see what pattern - if any - emerges that might help me predict the eleventh flip. I'm using a €2 coin, which has neither a head nor a tail. Luckily, it does have a harp on one side and a number 2 on the other: since they begin with the same letters as "Head" and "Tail", let's just go from there...

H H T H H T T T H T

Okay, that's five Hs and five Ts. Rats! Right, I'll do it again:

H H H H H T H T T H

That's better! Seven Hs and three Ts. Now, we can take two conclusions from this pattern:

Conclusion 1: There are more Hs, therefore H is more likely to come up next time because the coin seems to be "skewed" towards coming up H.

Conclusion 2: There are more Hs, therefore T is more likely to come up because it's overdue.

So let's see what happens: Here we go!

It's a T.

Hmm... Okay, we have our answer, but did the T come up because Conclusion 2 was correct, or was there another reason?

Well, the truth is very simple: the coin came up tails for no particular reason. That's what "random" means. Most people tend to forget one important thing about flipping coins: the coin doesn't know what came up on the last flip. So every flip might as well be the first one: the chances of the eleventh flip coming up tails is one in two.

But getting back to the question: can some numbers be luckier than others? Well, no, they can't. Like the coin, the numbers don't know stuff. They don't remember.

Gamblers make this mistake a lot: "Oh, the lottery wasn't won last week, that means it's more likely that it'll be won this week, so I have to play it." This kind of logic is only one step away from thinking, "if I don't play, I won't win. Therefore: if I do play, I will win."

Summing Up

We all take numbers for granted, and we all attribute arbitrary importance to them: thirteen is unlucky, seven is lucky, having the same birthday as a famous person is cosmically significant, it's exciting when the car's odometer comes up 99999.9, the world is going to end in the year 2000 simply because it's a round number (how foolish do all those people feel now, I wonder?), entering 5318008 into a calculator is somehow funny... I'll stop there.

Today's homework: find the first seven-digit prime number, and flip a coin that number of times. Write down whether each flip is Heads or Tails. When you're done, get a large sheet of graph paper lined with little squares, and, starting at the top left and moving right, fill in each H with a black pen, and each T with a red pen. When you get to the 314th square, move down to the left-most side of the next line and carry on until you are finished, tired, or out of ink.

You may be surprised at the pattern that emerges! Actually, you probably won't. But it'll keep you out of trouble.