Alice forgot the password to access to her bank account. She presses the button

Question

Alice forgot the password to access to her bank account. She presses the button ‘I forgot my password, and the system generates a temporary password for her to use. This password consists of 6 elements, where the first two are characters (namely, they can be any letter of the alphabet) and the last four are digits (namely, they can take any value in the set 10, 1,2, 3, 4,5,6,7,8,9. (a) What is the probability that Alice receives a temporary password with at least one 'A'? (b) What is the probability that Alice gets a temporary password with two (c) What is the probability that Alice getsa (d) What is the probability that Alice gets a temporary password with the (e) Assume that the third digit of the temporary password is a 5. What is the distinct characters? year of birth in the four digits? third digit that is strictly smaller than the fourth digit? probability that the second digit is strictly smaller than the third one?

Explanation / Answer

a) P(atleat 1 A) = 1-P(No A) = 1-(25/26)^2

P(atleat 1 A) = 1-0.925 = 0.075

b) P(distinct characters) = (26/26) * (25/26)

P(distinct characters) = 0.9615

c) P(year of birth) = (1/10)4 = 0.0001

d) There are 45 instances where 3rd digit is smaller that 4th digit

and there are total 10 possibilities

P(3rd is smaller than 4th) = 0.45

e) There are 5 instances that are possible for this scenario

P(2nd is smaller than third) = 5 / 10 = 0.5

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