The case of the random telephone numbers

An unusual pattern of telephone numbers recorded on suspect credit‐card purchases led to the development of a generalized multinomial distribution. One usage of credit cards included 26 telephone numbers with a pattern of 7 unique numbers: 16 of one kind, 4 of another, 2 of another, and 4 different ones. With the assumption that the probability of any digit appearing in a telephone number is 1/10, the pattern of the 26 telephone numbers is found to have a probability of occurrence on the order of 1/10 65 . That is, the chance of such a pattern occurring at random is roughly equivalent to winning nine times in a row the popular ottery, LOTO 6/49.