05AB1E, 27 26 bytes

Saved a byte thanks to Adnan.

5Å0₄FDŽH|2ôI‹O·ÝΩ©è>®ǝ]ε0Û

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Explanation

5Å0                          # initialize with a list of 5 zeroes
   ₄F                        # 1000 times do:
     D                       # duplicate the list
      ŽH|2ôI‹                # check if the input is larger than 44 and/or 59
             O·              # sum and double, yielding (0,2 or 4)
              ÝΩ             # pick a random number between and 0 and the number above
                ©è           # get the count in that level
                  >          # increment it
                   ®ǝ        # insert it at the same position
                      ]      # end loop
                       ε0Û   # remove leading zeroes on each level

05AB1E, 27 26 25 bytes

Saved a byte thanks to Adnan.
Saved another byte thanks to Kevin Cruijssen.

5Å0₄FD„,;ÇI‹O·ÝΩ©è>®ǝ]ε0Û

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Explanation

5Å0                         # initialize with a list of 5 zeroes
   ₄F                       # 1000 times do:
     D                      # duplicate the list
      „,;ÇI‹                # check if the input is larger than 44 and/or 59
            O·              # sum and double, yielding (0,2 or 4)
             ÝΩ             # pick a random number between and 0 and the number above
               ©è           # get the count in that level
                 >          # increment it
                  ®ǝ        # insert it at the same position
                     ]      # end loop
                      ε0Û   # remove leading zeroes on each level

05AB1E, 27 26 bytes

Saved a byte thanks to Adnan.

5Å0₄FDŽH|2ôI‹O·ÝΩ©è>®ǝ]ε0Û

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Explanation

5Å0                          # initialize with a list of 5 zeroes
   ₄F                        # 1000 times do:
     D                       # duplicate the list
      ŽH|2ôI‹                # check if the input is smaller than 44 and 59
             O·              # sum and double, yielding (0,2 or 4)
              ÝΩ             # pick a random number between and 0 and the number above
                ©è           # get the count in that level
                  >          # increment it
                   ®ǝ        # insert it at the same position
                      ]      # end loop
                       ε0Û   # remove leading zeroes on each level

05AB1E, 27 26 bytes

Saved a byte thanks to Adnan.

5Å0₄FDŽH|2ôI‹O·ÝΩ©è>®ǝ]ε0Û

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Explanation

5Å0                          # initialize with a list of 5 zeroes
   ₄F                        # 1000 times do:
     D                       # duplicate the list
      ŽH|2ôI‹                # check if the input is larger than 44 and/or 59
             O·              # sum and double, yielding (0,2 or 4)
              ÝΩ             # pick a random number between and 0 and the number above
                ©è           # get the count in that level
                  >          # increment it
                   ®ǝ        # insert it at the same position
                      ]      # end loop
                       ε0Û   # remove leading zeroes on each level

05AB1E, 27 bytes

5Å0₄FD44 59‚I‹O·ÝΩ©è>®ǝ]ε0Û

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Explanation

5Å0                           # initialize with a list of 5 zeroes
   ₄F                         # 1000 times do:
     D                        # duplicate the list
      44 59‚I‹                # check if the input is smaller than 44 and 59
              O·              # sum and double, yielding (0,2 or 4)
               ÝΩ             # pick a random number between and 0 and the number above
                 ©è           # get the count in that level
                   >          # increment it
                    ®ǝ        # insert it at the same position
                       ]      # end loop
                        ε0Û   # remove leading zeroes on each level

05AB1E, 27 26 bytes

Saved a byte thanks to Adnan.

5Å0₄FDŽH|2ôI‹O·ÝΩ©è>®ǝ]ε0Û

Try it online!

Explanation

5Å0                          # initialize with a list of 5 zeroes
   ₄F                        # 1000 times do:
     D                       # duplicate the list
      ŽH|2ôI‹                # check if the input is smaller than 44 and 59
             O·              # sum and double, yielding (0,2 or 4)
              ÝΩ             # pick a random number between and 0 and the number above
                ©è           # get the count in that level
                  >          # increment it
                   ®ǝ        # insert it at the same position
                      ]      # end loop
                       ε0Û   # remove leading zeroes on each level