05AB1E, 4 bytes

AƶJΩ

Beats all other answers. Try it online!

AƶJΩ  # full program
   Ω  # random character from...
  J   # joined...
A     # "abcdefghijklmnopqrstuvwxyz"...
 ƶ    # with each character repeated 1-indexed index of character in string times
      # implicit output

AƶJ generates the string 'abbcccddddeeeeeffffffggggggghhhhhhhhiiiiiiiiijjjjjjjjjjkkkkkkkkkkkllllllllllllmmmmmmmmmmmmmnnnnnnnnnnnnnnoooooooooooooooppppppppppppppppqqqqqqqqqqqqqqqqqrrrrrrrrrrrrrrrrrrsssssssssssssssssssttttttttttttttttttttuuuuuuuuuuuuuuuuuuuuuvvvvvvvvvvvvvvvvvvvvvvwwwwwwwwwwwwwwwwwwwwwwwxxxxxxxxxxxxxxxxxxxxxxxxyyyyyyyyyyyyyyyyyyyyyyyyyzzzzzzzzzzzzzzzzzzzzzzzzzz', so the chances are as follows:

a: 0.285%
b: 0.57%
c: 0.855%
d: 1.14%
e: 1.425%
f: 1.709%
g: 1.994%
h: 2.279%
i: 2.564%
j: 2.849%
k: 3.134%
l: 3.419%
m: 3.704%
n: 3.989%
o: 4.274%
p: 4.558%
q: 4.843%
r: 5.128%
s: 5.413%
t: 5.698%
u: 5.983%
v: 6.268%
w: 6.553%
x: 6.838%
y: 7.123%
z: 7.407%

Alternative, 4 bytes

AηJΩ

Try it online!

AηJΩ  # full program
   Ω  # random character from...
  J   # joined...
 η    # prefixes of...
A     # "abcdefghijklmnopqrstuvwxyz"
      # implicit output

AηJ generates the string 'aababcabcdabcdeabcdefabcdefgabcdefghabcdefghiabcdefghijabcdefghijkabcdefghijklabcdefghijklmabcdefghijklmnabcdefghijklmnoabcdefghijklmnopabcdefghijklmnopqabcdefghijklmnopqrabcdefghijklmnopqrsabcdefghijklmnopqrstabcdefghijklmnopqrstuabcdefghijklmnopqrstuvabcdefghijklmnopqrstuvwabcdefghijklmnopqrstuvwxabcdefghijklmnopqrstuvwxyabcdefghijklmnopqrstuvwxyz', so the chances are as follows:

a: 7.407%
b: 7.123%
c: 6.838%
d: 6.553%
e: 6.268%
f: 5.983%
g: 5.698%
h: 5.413%
i: 5.128%
j: 4.843%
k: 4.558%
l: 4.274%
m: 3.989%
n: 3.704%
o: 3.419%
p: 3.134%
q: 2.849%
r: 2.564%
s: 2.279%
t: 1.994%
u: 1.709%
v: 1.425%
w: 1.14%
x: 0.855%
y: 0.57%
z: 0.285%