5 keys and lock

Question

Of the five available keys, only one fits. Assign a random distribution of X to the number of times the lock is unlocked, if the key tested once does not participate in the following tests, and find E(X). (Sorry if you find the mistakes in content. Im from Eastern Europe.)

Answer 1

If unsuccessful keys are eliminated then it can take at most $5$ attempts to open the door with the following probabilities: $$P(X=1)=\frac{1}{5}$$ $$P(X=2)=\frac{4}{5}\cdot\frac{1}{4}=\frac{1}{5}$$

$$P(X=3)=\frac{4}{5}\cdot\frac{3}{4}\cdot\frac{1}{3}=\frac{1}{5}$$ Similarly, $$P(X=4)=\frac{1}{5}$$ $$P(X=5)=\frac{1}{5}$$

Therefore, $E(X)=1.\frac{1}{5}+2.\frac{1}{5}+\cdots+5.\frac{1}{5}=3$