### Instruction: Two jokers are added to a $52$ card deck and the entire stack of $54$ cards is shuffled randomly.  What is the expected number of cards that will be strictly between the two jokers?### Response: I need to find the average distance between the two jokers in the shuffled deck.
I can think of the deck as a circular arrangement of cards, since the top and bottom cards are also adjacent.
Then the problem is equivalent to asking: if I choose two random points on a circle, what is the expected length of the shorter arc between them?
I can use some geometry and probability to answer this question.
If I fix one point on the circle, say $A$, then the other point, $B$, is equally likely to be anywhere else on the circle.
The length of the shorter arc between $A$ and $B$ depends on the angle $\theta$ between them, measured in radians.
The angle $\theta$ is uniformly distributed between $0$ and $\pi$, since $B$ is equally likely to be in either half of the circle.
The length of the shorter arc is $r\theta$, where $r$ is the radius of the circle.
I can assume the radius is $1$, since it does not affect the expected value.
So the expected length of the shorter arc is the average value of $r\theta$ over the interval $[0,\pi]$.
I can use calculus to find this average value.
It is given by the integral of $r\theta$ divided by the length of the interval.
That is, $\frac{1}{\pi}\int_0^\pi \theta d\theta$.
I can evaluate this integral by using the power rule.
It is $\frac{1}{\pi} \left[\frac{\theta^2}{2}\right]_0^\pi = \frac{\pi}{2}$.
This means the expected length of the shorter arc between $A$ and $B$ is $\frac{\pi}{2}$.
But this is not the final answer, because I need to convert it back to the number of cards in the deck.
To do this, I need to know the length of one card in terms of the circle.
The circumference of the circle is $2\pi r = 2\pi$, since $r = 1$.
The length of one card is the fraction of the circumference that it occupies.
There are $54$ cards in the deck, so each card occupies $\frac{1}{54}$ of the circumference.
That is, the length of one card is $\frac{2\pi}{54}$.
To get the number of cards between $A$ and $B$, I need to divide the length of the arc by the length of one card.
That is, $\frac{r\theta}{\frac{2\pi}{54}} = \frac{27\theta}{\pi}$.
The expected number of cards between $A$ and $B$ is the average value of $\frac{27\theta}{\pi}$ over the interval $[0,\pi]$.
This is the same as multiplying the average value of $\theta$ over the interval by $\frac{27}{\pi}$.
I already found that the average value of $\theta$ over the interval is $\frac{\pi}{2}$.
So the expected number of cards between $A$ and $B$ is $\frac{27}{\pi} \cdot \frac{\pi}{2} = \frac{27}{2}$.
This is the final answer.