### Can you list the numbers?

A list of twelve numbers have a range of eight, a mean, median & mode of five, if you picked one at random you had a fifty percent chance of a prime number, & there are as many even numbers as odd. List all twelve members.

-/|\-

###
Solution to follow …

We have twelve numbers, so $n=12$ & the mean is
$$\stackrel{-}{x}=\frac{\Sigma {x}_{i}}{12}=5$$
$$\Sigma {x}_{i}=60$$
The median is five too but as there are an even number of elements the mode is the mean of the sixth & seventh element when listed in order, so
$$\mathrm{median}=\frac{{x}_{6}+{x}_{7}}{2}=5$$
$${x}_{6}+{x}_{7}=10$$
Furthermore, the mode is also five
???
The range is eight, so
$${x}_{\mathrm{max}}-{x}_{\mathrm{min}}=8$$
If the numbers are ordered this becomes
$${x}_{12}-{x}_{1}=8$$
The mode is five, so there are at least two fives in the set.

Knowing the
$P(X=\mathrm{prime})=\mathrm{0.5}$
so there must be six prime numbers (some are repeated, eg. five)

-oOo-