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# can anyone answer this probability question?

Sat 2 Jun, 2012 04:34 pm
System A uses 4-digit personal secret numbers (i.e., 0000{9999), and
on each login attempt it randomly chooses two of the numbers and asks
the user to cite them.
if attacker has made one observation of the credentials sarah used to successfully log into her account, and now he tries
to log in by himself and is faced with a fresh request of two digits.
Assume that he enters the requested digit if he knows it, and
otherwise makes a random guess. What is the probability that he
can enter them correctly?
if attacker has made two observations of the correct credentials.
What is the probability that he now knows all four digits of Alice's
PIN?
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markr

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Sun 3 Jun, 2012 09:24 pm
@mah1,
Part 1
There are C(4,2)=6 ways to request two digits.
Let's say the 4-digit number is ABCD, and let's say the attacker saw AB.
The possible requests the attacker would see are:
AB - 1/6 * 1 (he doesn't have to guess)
AC - 1/6 * 1/10 (he's got a 1/10 chance of getting the digit he didn't see)
AD - 1/6 * 1/10 (ditto)
BC - 1/6 * 1/10 (ditto)
BD - 1/6 * 1/10 (ditto)
CD - 1/6 * 1/100 ( he's got a 1/100 chance of getting both digits he didn't see)

Add them to get 47/200.

Part 2
Once he's made the first observation, there's only one observation (of the six that are possible) that will give him complete information. Therefore, the probability is 1/6 that he'll know all four digits.
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