PMF of throwing a die 4 times We throw a fair 6-sided die independently four times and let $X$ denote the minimal value rolled.


*

*What is the probability that $X \ge 4$?

*Compute the PMF of $X$.

*Determine the mean and variance of $X$.
My attempt:


*

*$(1/2)^4$ because that means each of the $4$ rolls, you either get a $4, 5$ or $6$.

*for $X=1: (1/6)(6/6)(6/6)(6/6)$
for $X=2: (1/6)(5/6)(5/6)(5/6)$
for $X=3: (1/6)(4/6)(4/6)(4/6)$
for $X=4: (1/6)(3/6)(3/6)(3/6)$
for $X=5: (1/6)(2/6)(2/6)(2/6)$
for $X=6: (1/6)(1/6)(1/6)(1/6)$

*I can calculate this once I know I did the PMF correctly
Did I do (1) and (2) correctly?

1) is now correct after your edit based on Daniel's comment.
2) is incorrect since a) it doesn't sum to 1 and b) $X = 4, 5, 6$ doesn't sum to what you got in 1).
My approach to this question is to break it down into a bunch of simpler problems as such:
For $P(X=1)$, let's consider the $4$ dice as dice $1, 2, 3, 4$. There are $4$ possibilities, the result can have $1,2,3,$ or $4$ ones in it.


*

*$1$ one: ${4 \choose 1} (\frac{1}{6})^1(\frac{5}{6})^3$

*$2$ ones: ${4 \choose 2} (\frac{1}{6})^2(\frac{5}{6})^2$

*$3$ ones: ${4 \choose 3} (\frac{1}{6})^3(\frac{5}{6})^1$

*$4$ ones: ${4 \choose 4} (\frac{1}{6})^4(\frac{5}{6})^0 = (\frac{1}{6})^4$
The sum of the above will give you the $P(X=1)$.
Similarly, you can solve for the rest. Make sure that the end results sums to $1$ and $P(X=4) + P(X=5) + P(X=6) = (\frac{1}{2})^4$.
