Probability question about guessing 3 out of 4 correct numbers of a pass code AND 3 out of 6 possible CAP sensitive letters or numbers.

[coxie1102]

Hi, I'm revising for an uncapped resit in about 3 weeks and I've come across a exercise question that I'm confused about.


This is the question:

An online banking website requires its account holders to choose a four digit pin number, and a case-sensitive password made up of digits and letters (the letters can be upper or lower case). Suppose you choose your pin number and a six character password at random (for the password, each character has the same chance of being any digit or upper or lower case letter). When logging in, the website asks you to specify three digits from your pin number, and three characters from your password, all chosen randomly. If someone tries to log into your account, choosing the three digits and three characters at random, what is the probability that their first attempt is correct, gaining access to your account?

This is the most neat working attempt I have:


After about an hour of trying this I gave in and looked at the given solution, but I don't understand why the solution is what it is.

This is the given solution:

Define N to be the number of possible entries (the choice of three digits and three characters) to the website, so for example, one entry could be 1,0,4,a,H,7. For each digit, there are 10 possibilities, and for each password character, there are 26 + 26 + 10 = 62 possibilities. For three digits and three characters, we have N = 10³ ×62³ = 238328000 possibilities. There are N possible entries, and one of them will work, so the probability is 1/N = 1/238328000 .

I have underlined the bit that's confusing me, and it's where it says that "one of them will work". I agree that one of the solutions will work, but I also think that more than just one of the solutions will work.
For example:
If we say the correct pass code was: 1234
and we say the correct pass phrase was: 987aB1
Then one possible combination that you could enter to gain access to the bank website could be: 123 and 987. But then again you could also enter 124 and 987 or even 123 Ba7. and it's this that's confusing me.

Thanks in advance for any help

 

[Dr.Peterson]

I think the problem is poorly stated, so it is not clear what the rules are. That's partly because it's just not how passcodes work, so we have no intuition of how to interpret it.

It appears that "the website asks you to specify three digits from your pin number, ... all chosen randomly" doesn't mean, as I think you are taking it, that you enter three digits and they have to match some three digits of the passcode. Rather, they mean that they ask for, say, digits 1, 3, and 4, and you have to get those right. They pick the digits, not you. So it's really identical to asking for all of a 3-digit pass code, and likewise for the pass phrase.

At least, if we interpret it that way, their answer is right.

I hope the real test would not be so badly written. They may be trying to avoid demonstrating a problem that is as straightforward as the one they will really use.