Question

2. I have n keys, exactly one of which opens the door. I try them one by one at random independently, removing the key from the fob and setting it aside if it doesn't work. Let X be the number of keys I try until I open the door. Find E(X). A. (n-1)/2 B. (n + 1)/2 C. n/2 D. n + 1/2

help please

Answer 1

Let X be the number of keys tried until the door opens

For x = 1

$P(x) = \frac{1}{n}$

For x =2

$P(x) = \frac{n-1}{n} \times \frac{1}{n-1} =\frac{1}{n}$

For x =3

$P(x) = \frac{n-1}{n} \times \frac{n-2}{n-1} \times \frac{1}{n-2}=\frac{1}{n}$

So the probability is 1/n for all x

So

$E(x) = \sum_{x=1}^{n} x\times P(x)$

$(1\times \frac{1}{n}) + (2\times \frac{1}{n})+...+(n\times \frac{1}{n})$

$=\frac{n(n+1)}{2} \times \frac{1}{n} = \frac{n+1}{2}$

Answer (B)

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