document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1089658>Question 1089658</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Someone has stolen a bank card and wants to use it at an ATM. PINS are designed to contain four numbers (0 through 9). ATM machines are designed in a way that after the first three attempts to log in using a wrong PIN, a bank card stays with a machine.<BR>\n" );
document.write( "What is the probability that a thief will guess correctly the PIN at his third attempt? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #704060 by </B><A HREF=/tutors/aboutme.mpl?userid=math_helper><B>math_helper(2461)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_helper><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=math_helper><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=704060\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I am interpreting your question to ask what the probability is that the thief guesses the PIN precisely on the 3rd guess.\r<BR>\n" );
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document.write( "Since there are 10000 PINs (0000, 0001, …, 9999) and he/she presumably guessed two other unique PINs already, the probability of guessing correctly on the 3rd attempt is: <BR>\n" );
document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+highlight%281%2F9998%29+ ALIGN=MIDDLE  ALT=\"+highlight%281%2F9998%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > or about <b> 0.00010002</b>\r<BR>\n" );
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document.write( "As an aside/related note:<BR>\n" );
document.write( "If you wanted to know the probability that he/she guesses the PIN using <em>at most</em> 3 guesses, then that is<BR>\n" );
document.write( "1 - P(no guesses are correct) = 1-P(first guess is incorrect)*P(2nd is incorrect)*P(3rd is incorrect) <BR>\n" );
document.write( "= 1 - (9999/10000)(9998/9999)(9997/9998) <BR>\n" );
document.write( "= 1 - (9997/10000)<BR>\n" );
document.write( "= 3/10000 <BR>\n" );
document.write( "= <b>0.0003</b>\r<BR>\n" );
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document.write( "Although I calculated it as pick1 followed by pick2, then pick3, notice how the result is equivalent to 3 picks out of 10000 (if there were 10000 slips of paper, each with a unique PIN written on it, you could take 3 out all at once), as you'd expect intuitively.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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