An ATM personal identification number (PIN) consists of a four-digit sequence. (a) How many different possible PINs are there if there are no restrictions on the possible choice of digits? (b) According to a representative at the authors' local branch of Chase Bank, there are in fact restrictions on the choice of digits. The following choices are prohibited: (1) all four digits 26 1 Probability identical; (2) sequences of consecutive ascending or descending digits, such as 6543; (3) any sequence starting with 19 (birth years are too easy to guess). So if one of the PINS in (a) is randomly selected, what is the probability that it will be a legitimate PIN (i.e.. not be one of the prohibited sequences)? (c) Someone has stolen an ATM card and knows the first and last digits of the PIN are 8 and 1, respectively. He also knows about the restrictions described in (b). If he gets three chances to guess the middle two digits before the ATM "eats" the card, what is the probability the 1 gains access to the account? (d) Recalculate the probability in (c) if the first and last digits are 1 and 1.
