A base- three-digit number is selected at random. What is the probability that the base- representation and the base- representation of are both three-digit numerals

To calculate the probability that the base-9 representation and the base-11 representation of are both three-digit numerals, we need the following parameters:

The smallest 3-digit base 11 number in base 10
The largest 3-digit in base 9 in base 10 with 3 digits
The count of base 10 3-digit numbers

The smallest 3-digit base-11 number is 100

When converted to base 10, the equivalent number is 121

The largest 3-digit base-9 number is 888

When converted to base 10, the equivalent number is 728

So, the total number of possibilities is:

Possibilities = 728 - 121 + 1

Possibilities = 608

The count of base 10 3-digit numbers is

Count = 999 - 100 + 1

Count = 900

The required probability is:

P = 608/900

Evaluate

P = 0.6753

Hence, the probability that the base-9 representation and the base-11 representation of are both three-digit numerals is 0.6753