Answer to Question #119411 in Statistics and Probability for harry

Question

Paula has three keys on a key ring, just one of which opens the door to her
house. As she approaches the door, she selects one key after another, at
random without replacement, trying each key in the door until she finds the
correct key.
(i) Draw up a probability distribution table for X, the number of keys Paula
tries before she opens the door.
(ii) Calculate the expected number of keys that she will try before opening
the door.

Answer 1

Let X be the number of keys that Paula will try before opening. 

$$P(X=1)={1\over 3}$$ $$P(X=2)={2\over 3}\cdot{1\over 2}={1\over 3}$$

$$P(X=3)={2\over 3}\cdot{1\over 2}\cdot1={1\over 3}$$

(i) Draw up a probability distribution table for X, the number of keys Paula tries before she opens the door. 

$$\def\arraystretch{1.5} \begin{array}{c:c} x & 1 & 2 & 3 \\ \hline p(x) & 1/3 & 1/3 & 1/3 \end{array}$$

(ii) The expected number of keys that she will try before opening

the door

$$E(X)=1({1\over 3})+2({1\over 3})+3({1\over 3})=2$$