Is Counting Every Outcome Necessary for Solving Probability Problems?

[KingNothing]

Hi...we are doing this now in math. Our teacher insists on writing every possible outcome in a random experiment, which is proposterous. Why write out and literally count something that is fully calculable?

I can get some things pretty well. We'll have questions like "4 coins are flipped" with "what are the chances that there will be at least 2 heads"?

Well...we also have some with dice...like rolling two d6's, what are the chances that the sum of the two numbers will be 4-7 inclusive? I know what the answer is, but how would you get it with just math? I think it would invlolve factorials.

Please..tell me EVERYthing there is to know about calculating probabilities with a set of so many given outcomes, etc.

 

[HallsofIvy]

Originally posted by Decker
Hi...we are doing this now in math. Our teacher insists on writing every possible outcome in a random experiment, which is proposterous. Why write out and literally count something that is fully calculable?

I can get some things pretty well. We'll have questions like "4 coins are flipped" with "what are the chances that there will be at least 2 heads"?

Well...we also have some with dice...like rolling two d6's, what are the chances that the sum of the two numbers will be 4-7 inclusive? I know what the answer is, but how would you get it with just math? I think it would invlolve factorials.

Please..tell me EVERYthing there is to know about calculating probabilities with a set of so many given outcomes, etc.

Didn't you just tell us that you DON'T WANT to know "EVERYthing". Otherwise you would be quite happy to "write every possible outcome" and wouldn't consider it preposterous.

Yes, I understand what you are saying: It is preposterous that your teacher would expect you to do a lot of work in order to understand what's happening when it is so much easier to just memorize formulas (what you clearly mean by "just math") rather than actually having to learn mathematics.

 

[KingNothing]

No, you misunderstand. And your implication of my simplicity is just rude. I understand why algorithms and formulas work just fine. I know why 1-((5/6)^2) yields the same result as making a huge table displaying every combination and counting all the outcomes of rolling two dice that are one number or another.

That's why I want to know other ways to do them without any unnecessary work. Unnecessary work is bad when it doesn't help you at all. Please don't imply that I don't understand them.

To put it simply, I am at a point where I understand why formulas/calculations work, and I understand quite well 95% of the time. That's why I want to know ways of doing it like that. To save resources.

I don't know why you had to go off and make implications.

 

[Singularity]

You are quite right in thinking it has to do with factorials
So, instead of writing out all those nasty sample points, let's just use combinations. In your exapmle of tossing 4 coins : Let A denote the event of abtaining exactly 2 heads

$$C^4_2 = 4! / [(4-2)! * 2!]$$

You will find that most of the time combinations will be your best bet. Hope I'm not going to get flamed for posting this.