A guy with keys trynna open door [ E(X) and Var(X) problem]

Question

So a guy attempts to open his front door with his $n$ keys, but of course only one key is right. If he eliminates incorrect keys as he goes, so he never tries a key more than once, find the expected value and the variance of the number of attempts required to open the door.

X: random variable of the number of attempts to open door

$x \in {1,2,3, ... , n}$ and $p(x) = \frac{1}{(n+1)-x}$

1. Expected Value $$E(X)=\sum_{x=1}^n xp(x)$$ $$=\sum_{x=1}^n x\frac{1}{(n+1-x)}$$
   $$=\frac{-1}{6n(2n^2+3n-2)}$$

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Question: Where is my mistake?

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Update:

Ok, so it should be $$=\sum_{x=1}^n x\frac{1}{n}$$ because, if I consider previous the failures of previous attempts, $$=1\frac{1}{n} + 2(\frac{n-1}{n})(\frac{1}{n-1}) + 3(\frac{n-1}{n})(\frac{n-2}{n-1})(\frac{1}{n-2}) ... $$ $$=\frac{1}{n}+2(\frac{1}{n}) + 3(\frac{1}{n}) ... $$ $$=\sum_{x=1}^n x\frac{1}{n}$$ $$=(\frac{1}{n})(\frac{n(n+1)}{2})$$

Is this right?

Therefore, $$E(x) = \frac{n+1}{2} and$$ $$Var(X)=\frac{n^2-1}{12}$$

Answer 1

If the guy randomly sorts the keys and then tries them in the sorted order, the probability that the correct key is at position $x$ is $P(X=x)=\frac{1}{n}$. If you want to calculate the probability using the conditional probabilities for the first $x-1$ attempts to fail, you will of course receive the same result: $$P(X=x) = \cfrac{1}{n+1-x}\prod_{k=1}^{x-1}\cfrac{n-k}{n+1-k} = \cfrac{1}{n+1-x}\cfrac{\prod_{k=1}^{x-1}(n-k)}{\prod_{k=1}^{x-1}(n+1-k)}=\cfrac{\prod_{k=2}^{x}(n+1-k)}{\prod_{k=1}^{x}(n+1-k)} = \cfrac{1}{n}$$ Therefore, $$ E(X) = \sum_{x=1}^{n}x\frac{1}{n} = \frac{n(n+1)}{2}\cdot\frac{1}{n} = \frac{n+1}{2}$$ Maybe this helps?

Answer 2

The probability values $p(x)$ you have calculated are the probabilities that $X$ is at most $x$. The event $X=x$ is not independent of the event $X=y$ for $y< x$.

For any $x\in\{1,\dots,n\}$ we know that $X=x$ if and only if the guy chooses the wrong key for $x-1$ successive times, and then chooses the correct key at the $x$'th try. Now the probability that the first chosen key is correct is $1/{n}$. The second key has probability $1/(n-1)$, and in general $p(x)=\frac{1}{n-x+1}$ as you discovered. Notice though, that these are not the probabilities associated with the event $X=x$. However, we can use these probabilities to find the probability distribution $P$ of $X$. From the characterisation given in the beginning of the paragraph we thus have $$P(x)=\frac{1}{n-x+1}\prod_{i=1}^{x-1}\frac{n-i}{n-i+1}.$$ For $x=1$ we define the empty product to be $1$. With this probability distribution you can apply the usual calculations.