When we roll a single six-sided die, what is the probability of rolling 3 even and 1 odd number?
I want to solve this using $P(E) = \frac{|E|}{|S|}$ where $|E|$ is the event space, and $|S|$ is the sample space.
I can get the sample space: $|S| = 6^4$ because we are dealing with a six-sided die being rolled four times.
But how do you get the event space for this? I know getting an even number of one roll is $\frac{1}{2}$ and same with an odd, but would it maybe be $3^4 + 1^4$ because we want three even numbers and one odd number?
Best Answer
In how many ways can you put the $1$ odd die? $4$
In how many ways can you choose a even number in one trial? $3$ ( in three,instead?)
In how many ways can you choose a odd number in one trial? $3$
So the probability is $\frac{4\cdot 3^4}{6^4} = \frac{1}{4} $
In these cases, one good way to think it's to build a 'mental tree' of the sample space and to distinguish the branches in the event space.