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An introduction to a teacher's guide on probability.

OK, let’s really talk about problem #3 from last night’s homework. And for the moment, let’s ignore part (e), and go ahead and assume that 1 4 of all white people are children. Based on that assumption, you would expect roughly 75 white people, about 19 of whom are children, and about 9 of whom are boys. If you answered exactly 75 8 , or 9.375, that isn’t a crazy answer. Of course, you can’t actually have 9.375 white boys in a room. But that is actually the “expectation value” for such an experiment. If you have a thousand rooms with a hundred people each, the average number of white boys in each room will probably be 9.375.

In the formal language of probability, we would say that for any randomly chosen person in the U.S. in 2006, there is a 9.375% chance that this person will be a white boy. That’s what “percent” means: out of a hundred.

What percent of the people in this class, right now, are girls?

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If you roll a die, what is the percent chance that you will get an even number?

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OK, that’s easy enough. But we’re going to tweak it a bit. Obviously, there is nothing magical about the number 100. We could just as easily ask “How many out of a thousand?” or “How many out of 365?” But what turns out to be most convenient, mathematically, is to ask the question “How many out of 1?”

This is how we are going to work with probability numbers from here on out, so it is very important to understand this numbering system!

  • The probability of any event whatsoever, under any and all circumstances, is always between 0 and 1. A probability of –2, or a probability of 2, is meaningless.
  • A probability of 0 means “It cannot possibly happen.”
  • A probability of 1 means “It is guaranteed to happen.”
  • A probability of 1 4 means “It has a one in four chance of happening,” or “If you try this 100 times, it will probably happen 25 of them,” so it is the same as a 25% chance.

After you have said all that, you’re ready to hit them with the worksheet “Introduction to Probability.” It should only take 10 minutes (half of which is spent on #2a).

Then come back. Let’s go over #2 carefully.

2b. The probability is 1 16 . You can see this from the tree diagram, but how could we have figured it out without a drawing? The answer—we’ve discussed this before, and it is absolutely central—is by multiplying. 4 possibilities for the first die, times 4 possibilities for the second die, makes 16 possibilities for the combination.

But here’s another way we can look at that same multiplication. The probability of “3 on the first die” is 1 4 . The probability of “2 on the second die” is also 1 4 . So the probability of both these events happening is 1 4 × 1 4 , or 1 16 .

When you have two different independent events—that is, neither one has an effect on the other—the probability of both happening is the probability of the first one, times the probability of the second one.

The idea of “independent” events is crucial here, of course, and you have to stress it. But it’s also a fairly obvious point, and there is a real danger of making it sound more esoteric than it is. If you spend ten minutes discussing the word “independent” you may do more harm than good. Consider trying this instead. Tell that class that you’re looking into a big box full of bananas. One out of every four bananas in the box is green; the rest are yellow. Also, one out of every three bananas is stamped “Ship to California”; the rest say “ship to New York.” Finally, half the bananas are over four days old.

  • What is the probability that a given banana is green, and destined for New York? 1 4 × 2 3 1 6 . One out of every six bananas have both of these attributes. Or, to put it another way, a given randomly chosen banana has a 1 6 chance of having both attributes.
  • What is the probability that a given banana is green, and over four days old? Well, not much. Not 1 4 × 1 2 1 8 . Because in general, as a banana gets older, it turns from green to yellow. So being green, and being old, are not independent: one makes the other less likely.

Now, ask the class, in pairs, to come up with a similar scenario. (It should not involve fruit!) They should think of two events that are independent, and calculate the probability of both of them happening. Then they should think of two events that are not independent, and explain why the probability of both of them happening is not the product of their individual probabilities.


“Homework: The Multiplication Rule”

Going over this homework, of course you want to make sure that the last problem gets answered. With a little thought, it should be obvious to anyone that if P is the probability that something will occur, 1 P is the probability that it will not occur. If it happens 1 time out of 5, then it doesn’t happen 4 times out of 5. This can be memorized as a new rule, along with the multiplication rule, but it is easier to see why it works.

Questions & Answers

Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
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for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
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what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
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abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
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Source:  OpenStax, Advanced algebra ii: teacher's guide. OpenStax CNX. Aug 13, 2009 Download for free at http://cnx.org/content/col10687/1.3
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