ATM machines ask for a 4-digit passcode (i.e. PIN number) to verify your identit

Question

ATM machines ask for a 4-digit passcode (i.e. PIN number) to verify your identity to retrieve cash upon inserting your ATM card. If you fail to provide the correct PIN number three times in a row, the ATM machine will confiscate your card, with a suspicion that you are not the authorized user of the ATM card. Suppose someone had stolen your ATM card, took it to a nearby ATM machine, and tried to withdraw some money from your bank account. What is the probability of this thief succeeding in withdrawing your money by entering random PIN numbers within three attempts?

Explanation / Answer

We can consider this as binomial distribution with p=1/3 and n=3

We need to find P(x<=3)=ncxp^x(1-p)^n-x

so using online calculator we get P(x<=3)>0.9999

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