Question

A) What is the probability that a randomly generated 7-digit password has at least one repeating digit?

B) What is the probability that in a randomly generated 7-digit password, all digits are distinct and they occur in a strictly increasing order?

Answer 1

Total digits are 10 which are: 0,1,2,3,4,5,6,7,8,9.

• Total 7 digit passwords without any restrictions = 10^7 = 10000000

• Total 7-digit passwords with distinct digits = 10P7 = 10*9*8*7*6*5*4 = 604800

A) What is the probability that a randomly generated 7-digit password has at least one repeating digit?

=> P(at least one digit repeating) = 1 - P(No digits repeating) = 1 - (password with distinct digits / total passwords)

= 1 - 604800/10000000 = 0.93952

B) What is the probability that in a randomly generated 7-digit password, all digits are distinct and they occur in a strictly increasing order?

When all digits are different, then there is only one possible way to arrange them in order (from smallest to largest).

Hence, P(digits are strictly increasing when all digits are different) = 1 / Total passwords with different digits

= 1/604800 = 1.65 * 10^-6