Let X denote the largest number shown on four thrown dice, what is the probability of X being smaller or equal to x?

Question

How do I solve question (c)(i) in the picture down below? I understand what they’re asking but I have no clue how to obtain the general formula in terms of x that is given... Can somebody who understands this question give a logical explanation on how to arrive at this answer? (Please do include some text in your answer so it’s easy to follow :))

Answer 1

If $x$ is the maximum of the four dice

then $P(X \le x)$ is the probability the maximum is less than or equal to $x$

which is the probability each of the four dice show $x$ or less.

Since the probability a particular fair die shows $x$ or less is $\frac x 6$ for $x=1,2,\ldots,6$,

and the dice throws are independent,

the probability all four show $x$ or less is $P(X \le x)=\left(\frac x6\right)^4$.