Finding the expectations of a geometric style distribution under certain hypotheses

Question

In an investigation into animal behaviour, rats have to choose between five similar doors, one of which is 'correct'. Correct choice is rewarded by food and incorrect choice is punished by a slight electric shock. If an incorrect choice is made, the rat is returned to the starting point and chooses again, this continuing until the correct door is chosen. The random variable $X$ is the serial number of the trial on which the correct response is made, thus taking values $1,2,3,...$

Find the expectations of $X$ under each of the following hypotheses. Give your answers exactly (i.e. integers or fractions, but not decimals)

(b) At each trial the rat chooses with equal probability between doors which have not so far been tried, no choice ever being repeated.

Answer 1

b)

For $k=1,2,3,4,5$ let $\mu_{k}$ denote the expectation under condition that this test is done by using $k$ doors.

Then to be found is $\mu_{5}$ and we have the equalities:

$\mu_{1}=1$

$\mu_{k}=\frac{1}{k}1+\left(1-\frac{1}{k}\right)\left(1+\mu_{k-1}\right)=1+\left(1-\frac{1}{k}\right)\mu_{k-1}$ for $k=2,3,4,5$

These $5$ equalities enable us to find the $\mu_{k}$ hence also $\mu_5$.

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c)

We have the probabilities $P\left(X=1\right)=\frac{1}{5}$ and $P\left(X=k\right)=\frac{4}{5}\left(\frac{3}{4}\right)^{k-2}\frac{1}{4}$ for $k\geq2$.

These equalities enable us to find the expectation by means of: $$\sum_{k=1}^{\infty}kP\left(X=k\right)$$