All-Numeric Passwords. The technology consultancy DataGenetics published the online document PIN analysis. In addition to analyzing PIN numbers, passwords trends were examined. Seven million all-numeric passwords were collected and yielded the following estimate of the probability distribution of the number of digits used in an all-numeric password.

Digits Probability Digits Probability 4 0.492 9 0.029 5 0.092 10 0.015 6 0.178 11 0.004 7 0.073 12 0.003 8 0.113 13 0.001 On average, approximately how many digits would you expect an allnumeric password to have? Explain your answer.

Expected Value. As noted in Definition 5.4 on page 254, the mean of a random variable is also called its expected value. This terminology is especially useful in gambling, decision theory, and the insurance industry, as illustrated in Exercises 5.39–5.42.