Is the formula for average time spent an arbitrary concoction? 
Let there be two boxes A and B. The probability that a mouse moves from A to B is 0.05. What is the average time span after which the mouse moves from A to B?

The book mentions that the answer is $\frac{1}{0.05}=20$ minutes. I have a few questions regarding this. 


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*Where did we get this formula? It does sound plausible: for example, if the probability of going to $B$ is $0$, then the average time that the mouse takes to go to B= $\frac{1}{0}=\infty$. However, let the probability be $0.5$. Then the average time taken is 2 minutes. How is this intuitive? Why could it not have been 1 minute or 10 minutes? All that the probability being 0.5 tells us is that the mouse spends half the time in A and the other half in B.

*Why minutes? Why not seconds, or hours? Isn't selecting minutes as the unit of time involved fairly arbitrary?

You are right the problem should include the time between tries. As stated all you can say is that the mouse averages $20$ tries between moves from A to B. 
