The answer is 18.

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Let Albert's ticket price be $a$ , Bob's ticket price be $b$ , Casey's ticket price be $c$ , Dexter's ticket price be $d$ and Edward's ticket price be $e$ .

Now, we work under the assumption that the progressions follow the sequence of the names listed. Casey's ticket price is the median, therefore, when arranging all the ticket prices in ascending order, we get

$a,\;b,\;c,\;d,\;e$

We know that $a,\;b,\;c$ is an arithmetic progression. Given that Bob's ticket price is $3$ dollars less than Casey, we can say that the common difference, $d_1 = 3$ . We can say that

$b=c-3,\;a=c-6$

We also know that $c,\;d,\;e$ form an arithmetic progression. Now, let the common difference, $d_2=p$ . We can say that

$d=c+p,\;e=c+2p$

We want to find the difference between Casey's and Dexter's ticket prices. That would be $d-c$ , which is actually the value of $p$ .

The mean of the 5 ticket prices is the mean of Casey's and Dexter's ticket prices. Convert this into an equation:

$\dfrac{a+b+c+d+e}{5} = \dfrac{c+d}{2}$

Substitute $a,\;b,\;d$ and $e$ :

$\dfrac{c-6 + c-3 + c + c+ p + c+2p}{5} = \dfrac{c+c+p}{2}\\ 2(5c + 3p - 9) = 5(2c+p)\\ 10c + 6p - 18 = 10 c + 5p\\ p=\boxed{18}$