what is the probability of rolling a 5 sided die 5 times and getting a 1 any number of times I am so confused; wouldn't it just be 1/5(5)?
The probability of rolling a six-sided dice five times and getting a 1 at least one time is 5/6, but how would this work for a five-sided dice?
 A: Suppose the die has $S$ sides and is thrown $n$ times. At any given throw, the chance of getting a 1 is  $\frac{1}{S}$ and the chance of getting something else than 1 is $\frac{S-1}{S} = 1 - 1/S$. Let $E$ denote the event where at least a 1 is thrown after $n$ throws. The complement of $E$ is the event where every single throw is not a 1.
Hence
$$P(E) = 1- P(E^c) = 1 - (1-1/S)^n$$
Setting $S = 5$ and $n=5$, we get $P(E) = 1 - (1-1/5)^5 = 1 - (4/5)^5$
A: Your dice has $5$ possible outcomes, each with a probability of $\frac{1}{5}$. So on each throw $P(1) = \frac{1}{5}$. The probability of rolling the dice $5$ times and getting $1$ at least one time can be easily calculated if you notice that the probabilities fit a binomial distribution.
$P(x = 0, n=5, p=0.2) = \binom{5}{0}0.2^0(1-0.2)^5 = 0.8^5 = 0.32768$
This means the probability of not getting any $1$ is $0.327$. Therefore the probability of getting $1$ at least one time is $1-0.327 = 0.67232 \approx 67.2\%$
A: 
The probability of rolling a six-sided dice five times and getting a 1 at least one time is 5/6

Okay, so the probability of rolling a six-sided die seven times and getting a 1 at least one time is $7/6$?
But probabilities can't be larger than $1$.
You cannot add probabilities unless events are disjoint (i.e. can't both happen at once). Here, it can happen that you get a 1 on more than one roll. Your calculation is double counting sequences with more than one $1$ in them.
In the simple case where we roll the six-sided die twice, write out each possible pair of rolls explicitly. There are $36$ options. $10$ include exactly one 1. There is $1$ option where both rolls land on 1. So the probability of getting a $1$ at least once is $11/36\neq 2/6$.
This generalises to the inclusion-exclusion principle.
