Here's an example of either poor engineering of store shelves or of incorrect loading of those shelves.

Now let the crying commence..

There's a field of physics engineering called statics. It is concerned with the forces and torques acting on objects at rest. If you want to build a bridge, you have to understand statics.

Here's an example of either poor engineering of store shelves or of incorrect loading of those shelves.

Now let the crying commence..

I received the email below from a reader in response to last Sunday's post :

Dr. Vann Priest,Regarding your Oct. 23, 2011 Oenobareus blog post about "Ask Marilyn":Like everyone, Marilyn sometimes makes mistakes or writes ambiguously,but not this time.The reason is that, according to Marilyn's problem, the writing downof the series occurs AFTER the 20 die rolls. In addition, either (a)or (b) MUST represent the actual result of that 20 die rolls.Marilyn doesn't give the calculations, but (b) is vastly more likelyto have been the actually rolled series.If Marilyn's problem had said that the 20 die rolls occurs AFTERwriting down (a) and (b), then your probability calculations forrolling (b) would be correct. But then the probability of either(a) or (b) being the actually rolled series would be spectacularlyunlikely.Best regards,[name redacted]

I wish to thank this reader, because I it caused me to go back and more carefully consider probability theory. I wondered if knowing that Marilyn actually did write down the sequence of rolls somehow affects the probability.

I no longer own my probability and statistics book from my undergraduate days, so did a quick Google search and found "Introduction to Probability" by Grinstead and Snell. This text is published by the American Mathematical Society, is freely available, and may be distributed under the terms of the GNU Free Documentation License.

Knowledge can affect the probability. [See chapter 4 - 'Conditional Probability'] Back in 1990, Marilyn wrote about the Monty Hall problem. Many people, including me, were convinced Marilyn was wrong. It was only after I sat down and carefully considered the effect of knowing what was behind door #3 that I was convinced that Marilyn was indeed correct, and I learned a lesson about conditional probability. See Example 4.6, p. 136 for a good discussion of the Monty Hall problem.

Here is a conditional probability example from the text -

Example 4.1 An experiment consists of rolling a die once. Let X be the outcome. Let F be the event {X=6},and let E be the event {X>4}. We assign the distribution function m(ω) = 1/6 for ω = 1,2,...,6. Thus, P(F) = 1/6. Now suppose that the die is rolled and we are told that the event E has occurred. This leaves only two possible outcomes: 5 and 6. In the absence of any other information, we would still regard these outcomes to be equally likely, so the probability of F becomes 1/2, making P (F |E) = 1/2.

So someone rolled a die, told someone else that the outcome was greater than 4. So we now know the outcome was either a five or a six. Since both are equally likely, the probability is 1/2.

Now let's discuss last Sunday's "Ask Marilyn" column. My analysis of the probability of throwing the two sequences (11111111111111111111 or 66234441536125563152) is correct. Both sequences are equally likely to be thrown.

But what happens when Marilyn tells us one of them actually occurred? Nothing! Conditional probability deals with the probability of future events based on knowledge of past events. There are no future events in this situation.

Now allow me to further explain why the sequence 66234441536125563152 is the more likely one. I alluded to it in last Sunday's post. The reason is entropy.

Entropy is defined to be a measure of the number of possible arrangements. Each distinct arrangement is called a microstate. Each of the rolls 11111111111111111111 and 66234441536125563152 are microstates. Both as I have shown are equally likely; this is the fundamental assumption in statistical mechanics However, with a thermodynamic system (and a good analogy to thermodynamic systems is dice), we usually do not concern ourselves with which microstate the system has. Physicists are concerned with the macrostate of the system. The macrostate of the system is specified by some measurable parameters. From the Second Law of Thermodynamics, we can infer that the most likely macrostate is the one with the largest number of microstates.

An example is in order. Let's consider the air in your room. To be able to write down the microstate, we would have to know the position and velocity of each molecule, but to write down the macrostate, we simply have to measure the temperature, air pressure, and the volume of the room. The air in your room fills the entire volume, because the number of microstates where the gas fills the entire room is larger than astronomical. The air does not occupy the bottom few inches, because the number of microstates, while still unbelieveably large, is tiny compared to when the air fills the room.

Here's the problem with the dice - it's too easy to emphasize either the microstates or the macrostate, and I'm convinced the confusion lies here. There are 3,656,158,440,062,976 ways (microstates) to throw a die twenty times. One of those must occur. Which one? We have no way to predict. It could be 11111111111111111111 or 66234441536125563152 or 56241113533264432213 or one of the other 3,656,158,440,062,973 possible rolls.

For throwing a die twenty times, the most likely thing (macrostate) to happen is for four numbers to come up three times and two numbers to come up four times, and this is what Marilyn describes as jumbled. For Marilyn's two choices, I pick 66234441536125563152, because this unlikely roll corresponds to the most likely macrostate.

After all this, I realize what really irks me about Marilyn's response to the math instructor. She makes no attempt to explain. All she writes in defense of her position is an appeal to her readers sense of what the correct answer is and a restatement that she is right. If I ever attempt to do in a classroom what she does in this column, my students may print this blog post out, fold it into a paper airplane, and bombard me. Just wait until my back is turned, so you don't poke my eye out.

Labels:
dice,
entropy,
Marilyn Vos Savant,
probability

In Parade magazine - that free magazine that appears in your Sunday paper - has a column written by Marilyn Vos Savant who is in the Guiness Book of World Records for supposedly having the world's highest IQ.

Here is today's column:

I’m a math instructor and I think you’re wrong about this question: “Say you plan to roll a die 20 times. Which result is more likely: (a) 11111111111111111111; or (b) 66234441536125563152?” You said they’re equally likely because both specify the number for each of the 20 tosses. I agree so far. However, you added, “But let’s say you rolled a die out of my view and then said the results were one of those series. Which is more likely? It’s (b) because the roll has already occurred. It was far more likely to have been that mix than a series of ones.” I disagree. Each of the results is equally likely—or unlikely. This is true even if you are not looking at the result.—George Alland, Woodbury, Minn.

Having the highest IQ does not make one immune from being wrong. The math teacher is right.My answer was correct. To convince doubting readers, I have, in fact, rolled a die 20 times and noted the result, digit by digit. It was either: (a) 11111111111111111111; or (b) 63335643331622221214.Do you still believe that the two series are equally likely to be what I rolled? Probably not! One of them is handwritten on a slip of paper in front of me, and I’m sure readers know that (b) was the result.The same goes for the first scenario: A person rolled a die out of my view and then informed me the result was one of these series: (a) 11111111111111111111; or (b) 66234441536125563152. It was far more likely to be (b), a jumble of numbers.

Here's why. The probability of throwing any number 1 through 6 on a fair die is 1/6. So throwing a 1 has a probability of 1/6. Throwing two 1s in a row is 1/6 x 1/6. Throwing three 1s is 1/6 x 1/6 x 1/6. Etc. Throwing twenty 1s has a probability of 2.7 x 10^-16. Not very likely is it?

Let's look at the other sequence. The probability of throwing a 6 is 1/6. The probability of throwing a 6, and then another 6 is 1/6 x 1/6. The probability of throwing a 6, 6, and a 2 is 1/6 x 1/6 x 1/6. The probability of throwing a 6, 6, 2, and a 3 is 1/6 x 1/6 x 1/6 x 1/6. And so on. Then the probability of 66234441536125563152 is also 2.7 x 10^-16. It doesn't make any difference if she plans on rolling the dice or she does it out of sight.

So what's going on, because Marilyn's answer does make common sense, even if her mathematics is off. To understand what really makes the answer (b) requires some understanding of entropy - this is what she refers to as jumbled.

Look at the first sequence. It is twenty 1s. Now examine the second sequence except don't pay attention to the order. That roll has three 1s, three 2s, three 3s, three 4s, four 5s, four 6s. The probability of throwing three 1s, three 2s, three 3s, three 4s, four 5s, four 6s is 0.239. 24% That's pretty likely. The reason this is so likely is that there are a very large number of ways to throw three 1s, three 2s, three 3s, three 4s, four 5s, four 6s. There's only one way to throw twenty 1s.

Marilyn's first answer is wrong. Her second answer (the "jumble of numbers" is more likely) is correct, but she makes no attempt to explain. Not so smart, in my opinion.

Labels:
dice,
Marilyn Vos Savant,
probability

My post from earlier today got me to thinking. The best reason for wine?

It's the perfect combination of biology, chemistry and physics.

University Professor and Donald Bren Professor of Biological Sciences Francisco Ayala, a geneticist at UC-Irvine, gave the university $10 million. The gift is to show gratitude to both the university and his adopted country. He was born in Spain and came to the U.S. as a student.

My first question is - Where does a biology professor get $10 million?

I love the answer. He grows grapes in Central California. By the way, he loves Pinot Noir.

I have to reprint one of the questions/answers from the Register article.

Q. Is there a tendency in our society to mix up religion and evolution?A.I am afraid largely so. I think it is wonderful to teach the Bible, but not to pretend the Bible is an introductory textbook for biology or astronomy.We succeed in keeping these kinds of things out of the schools, but then the impact on the public at large is not as good as you would expect to have. In the last few weeks, two or three of the Republican presidential candidates have expressed skepticism about evolution. And yet, evolution is confirmed as much as any scientific theory, and better than most. Evolution is confirmed as well as (the idea that) the Earth goes around the sun, or that matter consists of atoms.It's a matter of scientific ignorance. It's a matter of religious ignorance; as you surely know, most religious authorities, most churches, are in favor of evolution. As, famously, an Anglican minister -- a theologian -- said, (evolution) appeared first as an enemy, and has turned out to be our best friend, because evolution can now explain all of these sorts of cruelties or mistakes that exist in the world of life.Let's start with a simple example. The human jaw is not large enough for all the teeth. So we have to pull wisdom teeth -- sometimes one, sometimes two, sometimes three, sometimes all four. An engineer who designed the human jaw would be fired. And yet here we are, saying that would have been designed by God.Much more extreme and much more serious is the human reproductive system. The human reproductive system is a mess. Twenty percent of pregnancies end in spontaneous abortion, or miscarriages, in the first two months, because the human reproductive system is so badly designed. They blame God for 20 million abortions per year; there are about 100 million births in the world a year.

You may have heard the someone say "Oh, I'm a visual learner." Or maybe you know someone who doesn't take notes in class, because they say they learn best by just listening. They may classify themselves as auditory learners.

There are educational researchers who buy into this idea of learning styles. One of the most popular notions out there is the VARK model. VARK stands for 'Visual, Auditory, Reading (and writing), and Kinesthetic.'

What evidence is there for this idea?

Evidently there isn't any. My colleague, Dr. Kevin Smith, whose Ph.D. is in cognitive neuroscience, writes an excellent blog utterly destroying the VARK model. Go visit Learning Styles Evidence.

Subscribe to:
Posts (Atom)