Assigning swimming tracks!?

Problem: 7 swimmers are participating in a swimming match; 3 Americans, 2 Germans, 1 Australien and 1 Dutch. The swimming tracks 1 till 7 are randomly assigned.

Two questions:

1) What is the chance that 1 of the Germans is swimming in a outer track (so that would mean 1 german in track 1 or track 7)

2) What is the chance that at least 1 of the non-Americans is swimming in a outer track?

Could somebody please tell me how to calculate this?

I think I should divide the number of succesful outcomes by the total number of outcomes.

Total number = 7! = 5040

But how do I find the number of succesful outcomes in this case...

Answers should be

1) 0.476

2) 0.857

Re: Assigning swimming tracks!?

Quote:

Originally Posted by

**timmeh** Problem: 7 swimmers are participating in a swimming match; 3 Americans, 2 Germans, 1 Australien and 1 Dutch. The swimming tracks 1 till 7 are randomly assigned. Two questions:

1) What is the chance that 1 of the Germans is swimming in a outer track (so that would mean 1 german in track 1 or track 7)

2) What is the chance that at least 1 of the non-Americans is swimming in a outer track?

Answers should be: 1) 0.476 2) 0.857

Are you translating these questions into English?

I ask because it took me several tries to get the given answer to question 1).

That answer is to this question: “What is the probability that **exactly one** German is in an outside lane”?

BUT the way the question is written it means *at least one*.

BTW: $\displaystyle \dfrac{4\cdot 5\cdot 5!}{7!}=0.47619$ WHY?

Re: Assigning swimming tracks!?

1) $\displaystyle \frac{\binom 2 1 \binom 5 1}{\binom 7 2}$

2) $\displaystyle 1 - \frac{\binom 3 2}{\binom 7 2}$

Re: Assigning swimming tracks!?

Ridley that seems to be correct. Could give me a more elaborate explanation? Especially 2)..?