قوانین لیست و شمارش و احتمالات

Listing vs Counting vs Probability Rules

So in this video, we’ll talk about listing versus counting versus formal probability rules. So far, what we’ve had in the video series, we have seen a variety of techniques for solving problems including listing, counting techniques and formal algebraic probability rules. How does a student know which to use in a particular problem.

So I wanna make clear, in some ways, this is the hardest part about probability right here. You can think of it this way, so far all the videos we’ve seen, it’s kinda as if we’ve built this tool box, we’ve put a lot of tools in this tool box. And, of course, it’s important to know how to use each one of those tools, but the real question, the hard question is.

Now you’re faced with a probability question, how do you know which tools to take out of the box? So this is a very difficult process again, this is probably the hardest thing about probability. So I’m gonna start giving some hints about how to approach this, what are some things you can do to make this choice easier.

So first of all, use the formal probe, formal algebraic rules if, number one, the problem gives you algebraic expressions, P of A equals something, P of B equals something. Well, they’re giving you algebra, then that is just a dead red give away. They want you to use the algebraic rule. So that one is really easy.

If the items concerned are things like coins, cards, dice, those very simple inanimate objects. Those are really, really good for algebraic rules. So, if a problem is about that, chances are very good, you’re gonna be using the algebraic rules. If the language of the problems uses mutually exclusive or independent, that’s another dead give away right there you see those words.

Use the algebraic rules that’s what you’ll be using. Even if they don’t use those words explicitly they may use those ideas they may describe. An idea this doesn’t have an effect on another. These two things can’t happen together. You have to be able to recognize those ideas as being these conditions.

And again, if these conditions show up, whether they’re mentioned by name or not, use the algebraic rules. Now, there was a previous video that we had on analyzing problems. That’s one where we talked more in depth about which of the algebraic rules to use, or how to dissect a problem. Now, let’s talk about some of the other techniques.

Use listing only if the full list is very short, fewer than ten, we want to, have to be very careful here, if there are any combinations involved. Combinations always make the number of choices go way, way up. Suppose we’re talking about five books, and we’re talking about putting the books in order. Well, right away that’s five factorial, which is 120.

You’re not going to list that 120 things. So it’s not whether the number of items is fewer than 10. It’s whether the number of choices, the number of possibilities that we’re concerned with and the problem is fewer than 10. And so this is something really important to appreciate. It is extremely rare that listing will be the way to arrive at the solution of a problem.

There’s gonna be almost no problem in the test that we can solve completely, purely by making a list. But, this is a very important thought. Listing a couple examples at the outset might help you determine what approach to use. So, as you’re trying to figure out how am I going to attack this problem, often it can be very helpful just to list a couple examples of some of the elements, some of the examples of things that are success.

Some of the examples of things that are not successes, to give you an idea of how you are going to set things up. So again, listing is not necessarily a solution technique, but it may be something that, as it were, help you crack the nut of the problem, and start it. Finally, use the counting techniques if the problem involves selections of several elements from a set with certain restrictions.

One element is picked, or one is not picked. Or these two people are sitting next to each other. Those are all dead giveaways. As soon as you’re starting to pick elements from a set and what’s the probability of picking this kind of set versus that kind of set? Right there, you’re talking about counting techniques.

Don’t even bother dragging out the algebraic rules for that. It’s a counting techniques problem. Finally, now this is a tricky one. Whether you use the formal algebraic probability rules or the counting techniques, it may be easier to calculate or count the complement of what you want and then use the complement rule.

So this often takes a bit of creativity, a bit of thinking outside of the box to realize, oh wait a second, I could count or calculate all these things. But instead, I could just count this one thing or calculate this one thing, that is the complement, and that would be a much easier calculation. So again, that often takes some creativity, as you work with probability problems, you may start to see we can use the shortcut.

So in this video, I will give you some very broad clues for problem-solving. The most important thing I will say is the following. Figuring out how to approach probability problems. This is a right brain process, that is to say it is a pattern matching process. It is not a left brain process. Now, a left brain process I can just give you step by step rules and then explain very clearly.

That’s a left brain processes. Right brain processes are essentially non-linear, anything about pattern matching in the mind that’s a non-linear process. So there’s no way I can’t just give you a complete list of rules to follow to do this. The way you learn to do this is by doing problem after problem, and it is crucially important to read the solution sets of problems, even if you get the problem right, read the solution set to make sure that the way that it was solved in the solutions is the same way you’re thinking about it.

What you’re really looking for in the solutions. You’re not just looking for what’s the answer, it’s much more important to look at, how did they set up the problem? What were their first few steps? That’s what your looking for, you’re studying that very carefully in probability problems because you’re trying to build the pattern matching skills of your brain.

That takes time, but that’s ultimately what will help you in probability questions.