Remark: in the solution, the word Combination is used in the daily usage of the word, not in the combinatorial (mathematical) language.

Let the available 4 base colors be

,

the available 6 accent colors be

,

and the available 2 fonts for the embroidery be

,

so we have 4 b's, 6a's and 2 e's.

The event that Ray and Lilly design identical shoes can happen in the following way:

Ray has in total 4*6*2=48 possible combinations, and in each of these combinations, Lilly makes the same exact combinations of the 3- b's, a's and e's.

The sample space is 48 possible combinations of Ray * 48 possible combinations of Lilly = 48*48

(among these there are 48 matches of the combinations)

Thus P(designing identical shoes)=48/(48*48)=1/48=0.02

Answer:0.02