probability redux

16. If j is chosen at random from the set {4, 5, 6} and k is chosen at random from the set {10, 11, 12}, what is the probability that the product of j and k is divisible by 5?

SAT practice test

I missed this one.

VR on using backgammon to teach probability

VR wrote:

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.

Great minds think alike!

At some point, looking for self-teaching resources on probability, it came to me that I ought to stop trawling Amazon for introductory books on probability and start looking for introductory books on  card-playing.

Or card counting - ?

fyi: A lot of the Dummies books are terrific, I think. The book on public relations is very good, and I've just bought Grammar for Dummies because Stanley Fish said he likes it.

A few years back, I had a scheme to purchase all of the Dummies books and dedicate an entire bookshelf to them, the way people used to buy World Book Encyclopedias and give them pride of place.

Allison on why there is relatively little help available for people self-teaching probability

I believe there are several reasons why there is so little help for discrete probability at this [beginning] 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: Schaum's Outline of Theory and Problems of Probability (2nd edition)

instructional practice

I was looking for an old ktm post about Kumon this morning, and when I found it I rediscovered the concept of instructional practice, meaning practice that teaches.

I think that's probably what the Arlington Algebra Project's sections on probability provides.

question

Are there resources you like for teaching oneself beginning probability?

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!

decision trees from lsquared

from Lsquared:

If you're still thinking about this problem, you might find this page useful. It's not an explanation, per se, but there's lots of examples if you just follow the links, and it's intended for students who are almost certainly less mathematically sophisticated than you are.

This looks fantastic - thank you!

fyi: I mentioned Instapaper the other day. I've just stored the probability link in instapaper, where I'll be able to find it faster than I would find it here.

At least, I'll be able to find it faster at Instapaper until I max out Instapaper the same way I max out every other folder system I dream up.

help desk - probability

I'm having trouble with some problems in Mary Dolciani's Algebra and Trigonometry: Structure and Method Book 2.

p 749
11.  A bag contains 2 red, 4 yellow, and 6 blue marbles. Two marbles are drawn at random. Find the probability of each event.

a. Both are red.
b. Both are yellow.
c. Both are blue
d. One is red and one is yellow.
e. Neither is red.
f. Neither is blue.

For a, b, and c, I can solve this problem using two methods and arrive at the same answer:

a. Both are red.

2/12 x 1/11 = 1/66

OR, using the combination formula, nCr:

sample space (number of ways any 2 marbles can be picked out of 12):
12C2 = 12! ÷ 2!(12-2)!
          = 12 x 11 ÷ 2
          = 66

ways to pick 2 red out of 2 red:
2C2 = 1

probability of picking 2 red:
1 ÷ 66 or 1/66


Trouble is: when I use both approaches to solve d, I get 2 different answers, and I don't understand why.

d. One is red and one is yellow.

2/12 x 4/11 = 1/6 x 4/11 = 4/66 = 2/33

OR, using combination formula nCr:

12C2 = 66 (number of ways any 2 marbles can be drawn from 12)

2C1 = 2 (number of ways 1 red marble can be drawn from 2 red marbles)
4C1 = 4 (number of ways 1 blue marble can be drawn from 4 blue marbles)

so:

2 x 4 ÷ 66 = 4/33 (probability of drawing 1 red & 1 yellow)

2/33 ≠ 4/33

What am I missing here?

Thank you!

Barry on Singapore Math

The Singapore books do not go into what passes for data analysis and probability. I have the Singapore books and they are carefully structured and sequenced. Students have a very good grounding in fractions, adding, subtracting, multiplying and dividing, decimals, percents, ratios, some principles of geometry, and are given very good problem solving techniques using a device known as "bar modeling" which serves as a stalking horse to algebra. Algebraic manipulations are given in the sixth grade, but in the book I have they do not extend it to solving equations. Singapore teaches each topic area to mastery and then builds upon it. I've used it with my daughter and a friend of hers and had very good results. Others on this list have had similar experiences.

Barry Garelick

bonus post!

.

dropping in for a quick update

Just a quick update. We've finally gotten to Cartesian geometry -- after they did linear and quadratic equations (well, "did" in the current "let's mention it then move on to something else" sense), but only barely. The worksheet had a graph with little cartoon bubble labels: slope, x-axis, y-axis, intercept. The slope bubble pointed to the line, which could just as easily be the equation, but perhaps I'm being picky. Below it was a question: If x is 5, what will y be?

It would have been a perfectly reasonable question except that the tics on the axes weren't labeled. Were they 1, 2, 3, ..., or 2, 4, 6, ..., or 5, 10, 15, ... ? That little glitch made it just a tad difficult to answer the question.

That was presented along with set theory, which was your basic Venn diagrams, with unions and intersections.

Then they went back to "probability and statistics" (and yes, those are sneer quotes). I've already said I think it's bizarre to teach either in the 8th grade, but if you're going to teach it, then teach it. There was no new information presented the second time around. "Probability" was nothing more than your standard ball problem ("If there are 8 green balls and 4 red balls in the hat, what is the probability that you will select a red ball?"), which explains embarrassments like this. Worse was the "statistics" component, which was nothing more than median, mean, and mode.

I hate to break it to the math ed folks, but statistics is not "soft," and it is far more than measures of central tendency. Ultimately, even the hard sciences come down to statistics. Carbon-14 dating (and potassium-argon dating) are statistics. DNA testing is statistics. Epidemiology is statistics.

The math ed people I know wouldn't know a frequentist from a Bayesian, or MANOVA from a t-test. Call me cynical, but I can't help but wonder if that doesn't have something to do with this mess of a curriculum. Why revisit the same concepts over and over again? Couldn't they at least introduce -- in concept, if nothing else -- standard deviations or sample v. population?

If you're going to teach it, teach it. That's my outmoded, stale, dinosaurian view, anyway.

We did the "probability" and "statistics" worksheets in fifteen minutes. That's how much substance there was. But there were terms on the worksheet (she'd copied this one from somewhere, you could tell that) they hadn't covered (from the original source). She had told them to ignore anything they didn't understand (what kind of advice is that for a teacher to give a student?) but he wanted to know what they meant. So we talked about the normal distribution, standard error, and standard deviation (the terms from the original source on the handout).

(By the way, there's an interesting video of Peter Donnelly discussing common statistical errors here, if you're interested.)

In more general terms, I'm teaching Ricky formalism, to set up his problems in sequential, logical steps. He finds it anal retentive, and I'm not drilling him a lot because it frustrates him, but some every time I see him. He just doesn't understand why he should write "Let x equal the number of pears" at the top of the problem. I didn't either when I was his age, but I did it because I had no choice (that's the way math was taught back then). Now, I understand why, and that's why I'm passing it on to him. His biggest problem is that he's pretty good at figuring out how to solve a problem, and he doesn't see why he can't skip steps if he knows the intervening ones. I was like that. But there's a reason for it -- so he won't be like these students.

Anway, back to the beef and noodles.