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Unlucky Digit

Your unlucky digit is three (3). The school you will enter next year has purchased 300 new lockers that have been pre-numbered with 101 through 400. They will assign you a locker at random. What are your chances of avoiding the digit three (3) in your locker number?

Bonus: Your unlucky digit is instead 4. You will pick any positive integer at random. In what interval do you have less than a 0.5 probability to avoid any appearance of the digit 4 in the decimal representation? [Hint: logarithms might be helpful.]

source: variation on an idea in Keys to Infinity [Pickover].

Solution

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