The probability that a candidate comes with all $3$ pens having the same color Question

Candidates were asked to come to an interview with $3$ pens each.
Black, blue, green and red were the permitted pen colours that the candidate could bring.
The probability that a candidate comes with all $3$ pens having the same color is _________.

My Approach
$\binom{4}{1}$ to select 1 colors from $4$ available colors and then probability of each color $ =\frac{1}{4}$
Reqd probability$  =4 \times (\frac{1}{4} )^{3}$
is it correct?
We can write down a sample space: 

$$\Omega =\Big{\{}\{R,R,R\},\{R,R,B\}, \{R,R,G\},\{R,R,Y\},  \{R,Y,Y\}, \\ \{R,B,B\},  \{R,G,G\}, \{R,G,B\},  \{R,Y,B\}, \{R,Y,G\} , \\\{Y,Y,G\},  \{Y,Y,B\}, \{G,G,Y\},  \{G,G,B\}, \{B,B,G\},\\ \{B,G,Y\},\{B,B,Y\}, \{B,B,B\}, \{G,G,G\}, \{Y,Y,Y\}
\Big{\}}$$

So, if we assume that events are uniformly distiributed (since distribution is not mentioned in OP) we have: $$ P = {4\over |\Omega|} = {1\over 5}$$

Please, if somebody disagree with this solution then please write down your own  sample space and distribution and then tell me what is wrong.
We can write down a sample space: 

$$\Omega =\Big{\{}\{R,R,R\},\{R,R,B\}, \{R,R,G\},\{R,R,Y\},  \{R,Y,Y\}, \\ \{R,B,B\},  \{R,G,G\}, \{R,G,B\},  \{R,Y,B\}, \{R,Y,G\} , \\\{Y,Y,G\},  \{Y,Y,B\}, \{G,G,Y\},  \{G,G,B\}, \{B,B,G\},\\ \{B,G,Y\},\{B,B,Y\}, \{B,B,B\}, \{G,G,G\}, \{Y,Y,Y\}
\Big{\}}$$

So, if we assume that events are uniformly distiributed (since distribution is not mentioned in OP) we have: $$ P = {4\over |\Omega|} = {1\over 5}$$

Please, if somebody disagree with this solution then please write down your own  sample space and distribution and then tell me what is wrong.