Total cases $=4$ letter word is formed by using either $3$ identical letter and $1$ distinct letter or $2$ alike of one kind and $2$ alike of $2^{\mathrm{n}\mathrm{d}}$ kind or $2$ alike, $2$ distinct or $4$ distinct
Number of ways for $3$ alike and $1$ distinct $={{}_{}{}^{1}C}_1\times {{}_{}{}^{5}C}_1\times \frac{4!}{3!}=20$
Number of ways for $2$ alike and $2$ alike $={{}_{}{}^{2}C}_2\times \frac{4!}{2!2!}=6$
Number of ways for $2$ alike and $2$ distinct $={{}_{}{}^{2}C}_1\times {{}_{}{}^{5}C}_2\times \frac{4!}{2!}=240$
Number of ways for $4$ distinct $={{}_{}{}^{6}C}_4\times 4!=360$
The total number of ways $=20+6+240+360=626$
For favorable cases, the following possibilities are there (one D is always included)
Number of ways for $3$ alike $={}_{}{}^{1}C_1\times \frac{4!}{3!}=4$
Number of ways for $2$ alike $+1$ distinct $={}_{}{}^{1}C_1\times {}_{}{}^{4}C_1\times \frac{4!}{2!}=48$
Number of ways for $3$ distinct $={}_{}{}^{5}C_3\times 4!=240$
The total number of ways for favorable cases $=240+48+4=292$
Hence, the required probability $=\frac{292}{626}=\frac{146}{313}$