Someone has stolen an ATM card and knows the first and last

Someone has stolen an ATM card and knows the first and last digits are 3 and 9. What are the total number of possible values for the remaining two digits?

What is the probability that when the thief tries the card in an ATM that they will guess the pin by the third try?

Solution

from the remaining two digits the first can be anything from 0 to 9.hence 10 choices. similarly the 2nd one can also be anythig from 0 to 9. so total number of possible values for the remaining 2 digits are 10*10=100

let A be the event that he guesses the pin by third try

P[A]=P[guess the pin at first try]+P[fails to get at first but guess at 2nd try]+P[fails to guess at first two tries but guess at third try]=P[he chooses the correct combination from the 100 combinations]+P[ he choose at first try any one from 99 false combinations then at the second chosees the correct one]+P[ he chooses at first try one from 99 false combinations then at 2nd try from the remaining 98 false combinations then at third try chooses the right one]=1/100+99/100*1/99+99/100*98/99*1/98=0.03 [answer]