How Many Outcomes Result from Two Sequential Experiments?

[workerant]

Homework Statement

Two experiments are to be performed. The first can result in anyone of m possible outcomes. If the first experiment results in outcome number i, then the second experiment can result in any of ni possible outcomes, i=1 2, ..., m. What is the number of possible outcomes of the two experiments?

Homework Equations

The generalized form of the basic principle of counting comes to mind.

The Attempt at a Solution

Ok, I was thinking something like m*n^(i) or something along those lines.My prof said this is a very easy problem, so could it just be m*n or m*n*i or something? I'm a bit confused.

 

[buffordboy23]

One possible approach is to solve an analogous problem that matches the given data, and then try to use this result to solve the original problem. Let the first experiment be the flipping of a two-sided coin, with heads designated as 1 and tails designated as 2. If the result of the first experiment is a heads (1), then let the second experiment be the roll of a four-sided die (tetrahedron). If the result of the first experiment is a tails (2), then let the second experiment be the roll of an eight-sided die (octahedron). What is m, n, and i for this problem? What is the total number of possible outcomes of the two experiments?

Hopefully, this approach may be more enlightening for your learning than me giving you a simple formula.

 

[workerant]

m in this case would be 2 and n for (1) would be 4, while n for (2) would be 8, so I would say that there are a total of 4+8=12 possible outcomes for this particular experiment.

Now to bring this back to the original problem, I would think it could be just the sum of the n(i)s since for each result m, you are doing a different experiment so you don't have to multiply by m or anything. So would it just be to sum the nis?

 

[buffordboy23]

No. "n" is a constant (n = 4). It is "i" that varies from 1 to "m"; you said m = 2, so i = 1, 2, which is correct.

The rest of your thinking is correct too. The answer is a "sum". Can you express the answer with proper notation?

 

[workerant]

Well we only had one class so far and we didn't go over any proper notation (i.e. summations, which I think we do in the next class), we just did real basic problems which is why this question was a little more tricky than what we did in class.

Would it just be i(m+n) (i.e m is 2 and n is 4 and i is 2 in this case, giving 12 total)

 

[buffordboy23]

"i" varies according to the results of the first experiment. If i = 1, then experiment two has ni = (4)(1) outcomes. If i = 2, then the second experiment has ni = nm = (4)(2) = 8 outcomes as you have shown.

12 is the correct answer. You have offered a solution. Now comes an important step in the problem solving process. Can you verify that this answer is correct? Make a new problem: This time experiment one has 3 possible outcomes (1,2,3). Let's still keep n = 4 for simplicity. Therefore, experiment two will have (4,8,12) outcomes that are dependent upon the result of experiment one. Does the answer of this new problem support your answer to the original problem?

 

[workerant]

Ok I read a litle more in the book.

I believe now the answer to be the sum of the n(i)s from i=1 to i=m. This checks out.

If n(i)=n, then it would just be m*n.

Thanks for the help!

 

[buffordboy23]

I believe now the answer to be the sum of the n(i)s from i=1 to i=m. This checks out.

Yes. Exactly. Good job.

If n(i)=n, then it would just be m*n.

This is only true if i = 1, which implies that m = 1. Good.