How to calculate the probability of a die with a wild side?

Question

So I have a 6-sided die with 5 different values in 5 of their sides. Its sixth side can be treated as any of the other 5 values.

So my question revolves around which is the probability of getting any of the 5 values in the die? It can't be $\frac{1}{6}$, given one of their sides can add up to that probability. Also, it can't be $\frac{1}{3}$, because then it wouldn't add up to $1$.

So, what is the real probability of each value in this die?

Answer 1

Assuming that any one of the other 5 sides is equiprobable on the joker face, logically, it has to be $\frac{1}{5}$ since each side has an equal probability of being landed on, or picked on the wild face.

Now for the math. For any given fixed side, it has probability $\frac{1}{6}$ of being landed on, and $\frac{1}{6}\times\frac{1}{5} = \frac{1}{30}$ of being picked, assuming that the choice of fixed face is independent (and equiprobable) from landing on the joker face. Now: $$ \frac{1}{6} + \frac{1}{30} = \frac{5}{30} + \frac{1}{30} = \frac{6}{30} = \frac{1}{5} $$

Answer 2

Another way to look at it would be that the number you want has a $\frac{1}{3}$ chance of coming up and the other $4$ numbers each have a $\frac{1}{6}$ chance. That is, if there is a preferred number.