Abrazolica

Here's an interesting probability problem:

At a Christmas party Spike, Spud, and Sparky have three gifts to choose from. Two of the gifts are Yosemite Sam mud flaps and one is a bottle of Elvis Presley cologne. Spike really wants the cologne, Sparky wants the mud flaps, and Spud doesn't care what he gets. Spike knows Sparky wants the mud flaps and he also knows that Sparky helped wrap the gifts so he knows which ones have the mud flaps. Spike chooses his gift first, then Sparky chooses his and Spud gets the last one left. Just as they are getting ready to open the gifts, Spike has a brain tornado and asks to swap gifts with Spud. He thinks there is a higher probability that Spud has the gift with the Elvis cologne. Is he right?

And here's the answer:

At the last moment Spike realized that if his initial pick was mud flaps then Sparky picked the other set of mud flaps and Spud ended up with the cologne. This was the most likely scenario since Spike had a \(2/3\) probability of initially picking mud flaps and only a \(1/3\) probability of initially picking the cologne. It was therefore twice as likely that Spud had the cologne and not him. Asking Spud to switch gifts is the smart thing to do.

This problem is one of over 200 in our new book: Probability Problems and Solutions.

I read Musashi by Eiji Yoshikawa a couple of years ago and really enjoyed it. The book is about a sixteenth century Japanese Samurai named Miyamoto Musashi. His story has become a part of Japanese legend. The other day I found some quotes that I wrote down while reading the book. I think the quotes give the flavor of the book better than anything I could say so here they are. The book is a beautiful example of a man's struggle to break down barriers preventing him from understanding and experiencing true reality and from reaching his full potential. Everyone should read this book.

".. the human mouth is the gateway to catastrophe" (Musashi, p. 55)

"Sometimes people who are not quite right in the mind are taken by others to be geniuses" (Musashi, p.61)

"Do not attempt to oppose the way of the universe. But first make sure you know the way of the universe." (Musashi, p.663)

"Instead of wanting to be like this or that, make yourself into a silent, immovable giant. That's what the mountain is. Don't waste your time trying to impress people. If you become the sort of man people can respect, they'll respect you, without your doing anything." (Musashi, p. 680)

"It occurred to Musashi what an odd fact it was that most children could draw - and sing, for that matter - but that they forgot how to as they grew older. Perhaps the little bit of wisdom they acquired inhibited them. He himself was no exception. As a child he had often drawn pictures, this having been one of his favorite ways of overcoming loneliness. But from the age of thirteen or fourteen until he was past twenty, he gave up drawing almost entirely." (Musashi, p.839)

As in the case of other adults who have forgotten how to draw, his mind would work, but not his spirit. Intent upon drawing skillfully, he was unable to express himself naturally. (Musashi, p. 839)

Hyogo tried to rein in his emotions. Warriors had weak moments, foolish moments, like everybody else. Still, his duty, that of every samurai, was clear: to persevere until he reached a state of stoic balance. Once he had crossed the barrier of illusion, his soul would be light and free, his eyes open to the green willows around him, to every blade of grass. Love was not the only emotion capable of firing a samurai's heart. His was another world. In an age hungry for young men of talent, this was no time to be distracted by a flower along the wayside. What was important, as Hyogo saw it, was to be in the right place to ride the wave of the times. (Musashi, p.859-860)

"Musashi was kneeling silently, as though in meditation, his brush, ink box and brush pot beside him. He had already finished one painting—a heron beneath a willow tree. The paper before him now was still blank. He was considering what to draw. Or more exactly, quietly trying to put himself into the right frame of mind, for that was necessary before he could visualize the picture or know the technique he would employ.

"He saw the white paper as the great universe of nonexistence. A single stroke would give rise to existence within it. He could evoke rain or wind at will, but whatever he drew, his heart would remain in the painting forever. If his heart was tainted, the picture would be tainted; if his heart was listless, so would the picture be. If he attempted to make a show of his craftsmanship, it could not be concealed. Men's bodies fade away, but ink lives on. The image of his heart would continue to breathe after he himself was gone.

"He realized that his thoughts were holding him back. He was on the brink of entering the world of nonexistence, of letting his heart speak for itself, independent of his ego, free from the personal touch of his hand. He tried to be empty, waiting for that sublime state in which his heart could speak in unison with the universe, selfless and unhampered." (Musashi, p.958)

You can create the equivalent of a coin toss with no coins, no dice, or any other kind of randomizing device. All you need is three people. Have each person pick a positive integer of any size and write them down. To simulate the coin toss, you check to see if the sum of any two of the numbers is greater than the third. If it is, the toss is a head and if not, the toss is a tail. More precisely, if \(x\), \(y\), and \(z\) are the three numbers and \(x+y > z\), \(x+z > y\), \(y+z > x\), then the toss is heads otherwise the toss is tails. Amazingly this test will simulate a fair coin toss.

As an alternative to having a person pick a number you could just use the day of the month they were born. Since the numbers cannot be larger than 31 this method will not give you a perfectly fair coin toss but it will be extremely close to fair. When the numbers are limited to being no larger than \(n\) then the probability that the sum of any two numbers is greater than the third (coin comes up heads) is given by:

\[p(n)=\frac{1}{2}+\frac{1}{2n^2}\]

For \(n=31\) this differs from \(1/2\) by only \(.0005203\). If you use the birth month then \(n=12\) and the probability differs from \(1/2\) by \(.01020408\) which is still pretty close to being fair. The birth year will not work because the possible numbers must range from \(1\) to \(n\) and there's no one alive that was born in the year \(1\), I think. From the equation you can see that if there is no limit to how large the numbers can be, i.e. let \(n\) go to infinity, then the probability becomes exactly \(1/2\).

Now for a short sketch of why this works. Let \(f(n)\) be the number of ways to pick three integers between 1 and \(n\) so that the sum of any two is greater than the third, then \(f(n)-f(n-1)\) will be the number of ways where at least one of the integers is equal to \(n\). All three integers can equal \(n\) in only one way. Two of the integers can equal \(n\) either as \(nnx\), \(nxn\), or \(xnn\) where \(x\) ranges from 1 to \(n-1\) so the number of ways this can happen is \(3(n-1)\). One of the integers can equal \(n\) in 3 ways with the other two chosen from 1 to \(n-1\) in \(\binom{n-1}{2}\) ways, so the number of ways this can happen is \(3\binom{n-1}{2}\). The total number of ways where at least one of the integers equals \(n\) is then

\[f(n)-f(n-1)=1+3(n-1)+3\binom{n-1}{2}=1+3\binom{n}{2}\]

When \(n=1\) all three numbers must equal 1 and \(f(1)=1\). The above equation then gives \(f(2)=f(1)+4=5\). The five sets of integers in this case are (1,1,1), (1,2,2), (2,1,2), (2,2,1), and (2,2,2). Continuing on gives \(f(3)=f(2)+10=15\), \(f(4)=f(3)+19=34\), and so on. In general by iterating the equation you can see that \(f(n)\) for \(n\ge 2\) must be given by

\[f(n)=n+3\sum_{k=2}^n\binom{k}{2}\]

The sum in this equation reduces to \(\binom{n+1}{3}\) so that after simplification you are left with

\[f(n)=n(n^2+1)/2\]

To get the probability you divide this by \(n^3\) which is the total number of ways to pick three numbers from \(1\) to \(n\). This is where the formula for \(p(n)\) comes from.

This problem comes from a book of probability problems that we will be publishing soon.

What's going to happen tomorrow? The future is sending us signals about what is going to happen. The only problem is the information is being transmitted over an extremely noisy channel in a very intricately encoded form. So how do you pick up these signals and how do you decode the information contained in them. The signals are actually all around you in the form of evidence or data that you can collect right now. The big problem is that there is just too much of it, so you have to be selective. When you're selective then the process of decoding the signal becomes more error prone. Maybe with enough data you could decode the signal with no error, but probably not.

Even the laws of physics have moved beyond the idea that the future is perfectly predictable. Only classical mechanics has perfect predictability. The future of a particle in classical mechanics is completely encoded by a function called a Hamiltonian whose inputs are values that you can measure right now. But nature is fundamentally quantum mechanical. The future of a particle in quantum mechanics is also determined by something called a Hamiltonian but in this case it's an operator and not a function. The operator typically has many solutions called states and the particle's future, unless it is disturbed or measured, exists as a superposition of these states. In the future, when you look at the particle in some way, it will magically collapse into one of these states. It is tempting to say that the particle must have been in that state all along and if only you had some missing piece of information you could have predicted it. The problem is that experiments have shown that the particle really is in a superposition of states until you measure it.

So in physics there is no such thing as perfect predictability. A clockwork universe does not exist. Does this apply to what you're trying to predict? Maybe or maybe not. When we launch a spacecraft to the outer planets, its path can be predicted very accurately. Most prediction methods are based on the premise that the future is encoded by some function of things we can measure right now. Even something like a neural network is based on the idea that there is a function that determines the future. You can even define the equivalent of a Hamiltonian for some types of neural networks (see Hopfield Network for example).

It reminds me of Borges' The Library of Babel where he describes a fictional library that contains every possible book that could ever be written. This means that some book in the library contains a perfect description of what will happen tomorrow. The problem is the number of books is almost infinite and the vast majority are complete nonsense so finding the right book is virtually impossible. Some prediction methods implicitly assume there is a functional equivalent of The Library of Babel. This is a library containing every possible function. If you could just find the right function then the future would be perfectly predictable. The approach works surprisingly well for a lot of things.

But you should never be satisfied with a prediction method that says "this is what's going to happen". You want a prediction system that says "this, that, or the other" is going to happen and with these probabilities. In other words you should assume that the future is in a superposition of states. Some of those states are more probable than others but all are possible.