Someone (either lgm or lsquared, I think) recommended the Arlington Algebra Project, which looks like it's probably terrific for my purposes - though it lacks an answer key, which is not good.

I'll also be using this site (thank you!)

Anything else?

I wonder if I should take the AP statistics course at ALEKS.

**update**

kcab recommends The Art of Problem Solving, which turns out to have a course that sounds like exactly what I need: Introduction to Counting and Probability

Thank you!

## 24 comments:

What about Alcumus, at Art of Problem Solving? I'm only using it for fun, and to suck the kids into doing problems, but it seems like the combination of problems, answers, & video instruction could work. I think you might be able to set it so you only get the counting and probability problems.

I just had a flashback to my first day of my college Statistics course. He handed out the syllabus and said that there were, unfortunately, no Statistics tutors in the Math Lab because Statistics students forget everything they learned after they walk out of class on the last day.

Now, I had some pretty jaded profs in college, but I think he won the prize.

kcab - thank you!

Speaking of Art of Problem Solving, I found what looks like a terrific book of middle school contest problems, which was written by a former teacher who is now at Art of Problem Solving.

Here it is: Competition Math: for Middle School (Volume 1) by J. Batterson

Terri - It's amazing how little help there is on the web for beginning counting & probability compared to beginning algebra.

So far I haven't found the purplemath equivalent for probability.

The Cartoon Guide to Statistics is a good starting point for probability and statistics. The Art of Problem Solving book is more about combinatorics than about probability, though it is a very good intro for that.

I used

- Dolciani et al Pre-algebra: An Accelerated Course followed by

-Dolciani et al Algebra Structure & Method Book 1

for my afterschooling needs; will go on to the mat'l in the Alg. II book when the school course gets to it.

Those plus a solutions key or a tutor would be fairly low cost.

Khan Academy seems worthwhile too, although I haven't seen all the Probabiity mat'l...it's on my list to look at for this year's resource list.

I believe there are several reasons why there is so little help for discrete probability at this level:

1. It requires *significant* mathematical maturity to even begin to understand what's going on.

2. Almost no one understands probability anyway.

3. Even if you do understand, it is extremely difficult to be careful and precise enough to never make a mistake in describing the problems or wording the solutions.

4. Most of work in discrete probability rewards cleverness. Few methods for tutoring individuals, or supporting individuals in their own learning can teach cleverness.

I'll elaborate on these in another comment.

But this is where you should start: http://www.amazon.com/Schaums-Outline-Probability-Seymour-Lipschutz/dp/0070379823

re: 1: what I mean by mathematical maturity:

In order to calculate a probability for a discrete system, you need to understand the concepts of sample space, event, universe, independence, conditional dependence, labeling, ordering, union, intersection. This doesn't even cover understanding that probability theory *is not telling you about cause and effect*, and that the results about what happens on the ith event may have happened before or after in time from the i+1th. These are serious concepts, and the expectation that they will be clear without a lot of prior mathematical knowledge is probably in error.

2. Almost no one actually understands probability. That's why casinos and their forefathers have made money for hundreds, if not thousands of years. Human intuition is almost entirely wrong when trying to understand probability, which means you have to rely only on what the math tells you-- and that requires significant mathematical maturity. But that's not the only way in which people don't understand. Ask probabilists what a probability is, what it means, and their answer will be different than that of a frequentist, different from that of a bayesian, different from most physicists. Since none of them agree, it means we, generally speaking, don't really understand how to make it clear enough to teach youngsters.

I was looking at my son's math books (old Everyday Math to his current Algebra II book) on what they teach for probability. The early years deal with simple concepts and an approach that implies that with a few concepts and tools (decision tree), you can figure it all out. Then comes simple problems of whether events are dependent or independent. You can still get away with simple rules and logic to figure it out.

Then it changes. As Allison says, the problems become more clever. They are designed to make it difficult to apply logic and basic concepts of probability. I never had any problem with the formulas for combinations and permutations, but I did have problems understanding how to translate the words of the problem into a formula. Years ago, I bought Schaum's outline for Probability and Statistics and worked most of the problems. Even after that, I still felt that I could not solve any problem. I see that in my son's Algebra II book. You can look at the solved examples and everything looks easy, but there is no guaranteee that you will have a clue when see a new problem.

This is what I have to figure out how to teach my son, even though I still struggle with it myself.

I have been playing chess with my son for a few years. Chess is deterministic. In order to introduce him to probability concepts, I have started to play backgammon with him and bought him the "Backgammon for Dummies" book, which uses probability concepts to explain what is good play. Other games, such as blackjack and poker, can be used to teach probability. Many good games players can calculate probabilities well, even if they never read a book using the mathematical terminology Allison mentioned. Maybe books on those games can be used to teach probability in a palatable way.

This raises an interesting question. I'm looking for a direct path to understanding for my son. I want to reduce the process down to something that is almost rote. I want him to be able to quickly dissect all of the clever problems. His Algebra II book tries to deal with the issue of what the words mean, but this is limited to whether the events are dependent or independent.

Some problems fall into categories, like how many different arrangements of letters can the word MISSISSIPPI form. However, test makers seem to go out of their way to add in some sort of new spin to problems. I can't quite figure out whether this reflects an understanding of probability or language. (or the language of probability) Perhaps the difficulty lies in trying to translate words into separate events.

Steve,

I would suggest practicing rigorously how to turn the words into math with your son. Being able to tell right away which model you're working with is key.

So, you're given a problem like the one Catherine had about people in offices.

First question: what's the sample space? That is, what do individual solutions look like? The answer is each point in the sample space is a complete assignment of every person to some room. So that's not like a coin toss model.

Then second question: which probabilistic model does this fit? Answer: balls and bins. Balls and bins is where you care about individually labelled balls being assigned to individually labeled bins, but the order of the balls in bins doesn't matter. (because tossing balls into a bin is just a collection, there's no order in a bin.) Here, the people are labelled (all people have names, so all people are always labelled), and the rooms are labelled, but it didn't matter what order they were assigned to the room.

Third question: for balls and bins, how do you work probabilities?

For balls into bins, you do the factorials. you count the total ways that all people could be put into all available office spaces, then you count the ways with the characteristics you care about.

Other problems have different models. Some are coin tossing models--bernoulli process, it's called more formally. Tossing a die is like a coin toss problem, but with different values for p and q.

Steve,

Here are some things I'd try to drill into you and your son re: discrete probability. I find these are the fundamental errors that students make when trying to make sense of it, and it erodes their confidence in ever attacking problems.

One, a probability is a value, GIVEN OUR IGNORANCE, of how likely a specific outcome is.

It is FUNDAMENTAL that you impress this on your son.

OF COURSE you solve a physics problem to predict 100% perfectly the outcome of the toss of a fair coin, if you started with x,y,z, px, py, pz and all the needed impulses, air resistance effects, etc. There is NOTHING INTRINSICALLY random in a coin or a die.

We talk about coins and dice probabilistically from the standpoint of our ignorance: if we don't know the dynamics of the coin or die, and we don't know about prior outcomes, then, we have no way to prefer one outcome over the others. That's why the probability of a fair coin is 1/2 for heads on the ith toss--because we're not computing the classical mechanics to know how it works, and we can't infer anything from the prior tosses.

Note that if we don't know a coin is fair, and we see 100 heads in a row, we've got a much better than 50% chance that the next one is heads--because we're not as ignorant anymore.

I think this is highly confusing for most students. They think the rules of physics suddenly stopped applying, but don't know why. But probability isn't telling us about nature. It's telling us about what we don't know about nature.

--I want to reduce the process down to something that is almost rote.

I think the process can only be rote if the understanding is present. If we force them to become language detectives, then our kids make mistakes, lots of them, because they aren't trying to make sense of the problem.

That said, you want the process of breaking down the problem to become rote, yes: it should be rote to ask oneself 'what is the sample space?' 'does that tell me what model to use?' but looking for keywords to figure out the model is dangerous.

The test writers will be trying to cull those students from the herd.

So the process you want to imbed is to recognize that there are underlying models, and the problems fit them. as he gets more sophisticated, he'll get better at that mapping. it's better to spend time getting him used to the model than it is actually calculating the probability of an event until he's got those models down.

Allison:"

One, a probability is a value, GIVEN OUR IGNORANCE, of how likely a specific outcome is."For what it is worth, I like to approach this via a "coin in a box" approach. The basic idea is to flip a coin and place it into a box without looking at it. Most students shouldn't have a problem with the idea that the coin *IS* either heads or tails, but we just don't know which. Clearly, someone else could look into the box and then they would know, but for us it is still 50:50.

One can go from there: Would you (the student) bet on heads vs. tails with someone who had looked into the box? How about with someone who hadn't? Again, it should be obvious that we are measuring a certain amount of ignorance here. One can then go from this to odds *before* you flip the coin.

-Mark Roulo

Mark, it's great you do this. I think it's highly uncommon.

You'd be surprised how many textbooks and professors argue about whether a flipped coin that we've not looked at really has probability 1 or prob 0, not 1/2. There are many many statements by phd level folks who should know better that absolutely do not understand what the 1/2 represents. Physicists and economists might be worse than mathematicians, but even in mathematics, this mistake is common. (In fact, in physics, we have at least 5 generations of people who think special rules apply when you look at the coin in the box vs. you don't, and that it changes the fundamental nature of matter whether you look or not!)

But I don't think it's so obvious: if the outcome is known, then what does it mean that the probability of an outcome is 50/50? We talk about "the outcome", but our subjectiveness isn't in the question. No other mathematical statements we calculate are subjective like this. Sums, products, quotients, proofs are outcomes none of which are subject to who looked at the answer.

And worse, once teachers fail to tell their students that OF COURSE the physical process isn't random if you performed it properly, it seems like the subjectiveness is because of some bizarre metaphysical property of coins or dice, and then probability gets even weirder.

Glad to get the Schaum's recommendation - that's the book I chose today before seeing this.

I didn't buy it yet because I ordered the Art of Problem Solving book.

Given that I don't have a teacher, I'm thinking that a book geared to middle school students is what I need.

It'll be interesting to see whether a book geared to mathematically gifted middle school students just beginning probability works for a non-gifted, middle-aged beginner...

"it should be rote to ask oneself 'what is the sample space?'"

This is what I'm talking about. I've went back over my old marked up version of Schaum's and remembered many of my old questions. Then I went to the chapter on probability in my son's Algebra II book, and they had the same issues. My reaction is that both were written by those who forgot what it was like to learn the material in the first place. It doesn't have to be that confusing. The words and language have a lot to do with it. Take the word "event". It's too easy to think that this is something that happens, like a roll of the die. A conditional probability can refer to one roll of the die, but talk about two events.

I use the word "rote" in an ironic fashion. When you really understand a subject, the process almost becomes rote. I'm not talking only about some sort of word detective skill, but that is important.

"So the process you want to imbed is to recognize that there are underlying models, and the problems fit them."

And words lead you to those models. I noticed that both Schaum's and my son's Algebra II book do not help with this. One can easily do the problems in each section (on conditional probability, for example), but once the problems are all mixed up with variations, it becomes more difficult.

"I'm thinking that a book geared to middle school students is what I need."

My view is that the real understanding problems arise in algebra II.

"

And worse, once teachers fail to tell their students that OF COURSE the physical process isn't random if you performed it properly, it seems like the subjectiveness is because of some bizarre metaphysical property of coins or dice..."Some of this may be ignorance. Some of it may be smart people getting hung up on details that obscure the point of the thing they are trying to teach/explain.

Quantum Mechanics, for example, *really* does appear to be random/non-deterministic/whatever at a fundamental level. In a way that Newtonian Mechanics isn't. But for flipping coins, who cares?

Then we have the much larger percentage of teachers who may just not quite understand ... much like the K-8 teachers teaching fractions ... :-(

-Mark Roulo

Steve,

Have you spent time thinking about urn problems/balls in bins problems?

Books glibly throw around trivial examples of independence and conditional probability, but I can't imagine there is a single successful explanation in an algebra 2 book for why we can multiply the probabilities of successive pulls of a ball from an urn (or equivalently, successive tosses of balls in bins). When we do these formulations in factorial land, by counting all orderings, we ignore this detail. but if you try to go back and relate it to the probability of an event, suddenly we're multiply things like 1/k * 1/(k-1) for a series of events that are quite dramatically not independent.

The proof of why you can in fact multiply these probabilities requires understanding enough conditional probability to see that given what you've done, future pulls are independent of what you'e already done. Not exactly trivial.

Have you run up against this problem? It's a real sticking point for many kids who try to grok what's going on--why can they multiply the probabilities here?

--Quantum Mechanics, for example, *really* does appear to be random/non-deterministic/whatever at a fundamental level. In a way that Newtonian Mechanics isn't. But for flipping coins, who cares?

This is not the place for a discussion of what's wrong with QM, but there are many physicists who believe that QM does no such thing, and it's a fundamental misunderstanding of probability theory at the core that misled them to think this way. The work of ET Jaynes is probably the best place to start on this issue. for anyone wanting to discuss this more offline, email me.

Regarding The Cartoon Guide to Statistics, and in general learning statistics and probability, haven't we been here before?

"Have you run up against this problem?"

There is a disconnect somewhere. I don't spend much time with probability problems, so I've never had to really figure it out. Even with following the proofs in Schaum's, I start to lose a sense of what's going on. I guess now I will have to try to really figure it out.

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