Let there be n different pots arranged in a line, and n lids in a cupboard.
Each lid fits a single pot, and all the lids are different.
In the first round take a lid and put on the first pot in the line.
In the second round take a lid and put on the second pot in the line.
Continue until the $n_{th}$ lid- all lids are taken out at random and without replacement.
a) What is the probability of both the first and last lid fitting their corresponding pots?
b) N signifies the number of pot-lid pairs that fit each other after the process is done. Pick the right option:
- N is a random variable with Binomial distribution.
- N is a random variable with Geometric distribution.
- N is a random variable with Hypergeometric distribution.
- N is a random variable with an unnamed distribution.
- None of the above.
My thoughts:
a) There are (n-2)! different ways to arrange the lids in between the 1st and last pots.
So I think the answer would simply be $\frac{(n-2)!}{n!}$ = $\frac{1}{n*(n-1)}$, or simply a chance of $\frac{1}{n}$ to fit the 1st pot multiplied by $\frac{1}{(n-1)}$ for the last one.
b) Really not sure about the answer here-
I can say its not Binomial, because the chance of a lid fitting a pot is not a constant, which is also the reason it can't be Geometric. I think it might be Hypergeometric, but Im having trouble with fitting the parameters to this question...
Definitely haven't studied well enough to have anything smart to say about the last 2 options...
Any help would be greatly appreciated