Expert: Ahmed Salami - 8/9/2010

Question
QUESTION: I have a probability question.The question is two people meet.Both have bank cards with 4 pin numbers each.What is the probability of the first person having the last 4 digits of their telephone number to exactly match the second persons? and the second person having the exact last 4 digits of their telephone number as the first person's bank card pin number?
Thank you Ahmed

ANSWER: Hi Andrew,
Let the last 4 digits of the first person's phone number be ABCD and that of the second person be EFGH.
First, a little background. Suppose we both have single digits, what is the probability that they're the same? Its 1/10. This is so because from the 10 single digits i can have, only 1 matches yours. The probability that A matches E is thus 1/10.
The probability that (A matches E) and (B matches F) and (C matches G) and (D matches H) is therefore
(1/10).(1/10).(1/10).(1/10) = 1/10000
The probability of the second situation where the second person's last 4 phone number digits matches the first person's bank card pin can be argued in exactly the same way and you get the same value of 1/10000.
Now, the combined probability of both situations occurring simultaneously is simply
(1/10000).(1/10000) = 1/100000000

Looking at it in another way. Suppose there are three of us each with single digits. The number of possible combinations of digits is
(10)(10)(10) = 1000
Of this number, there are only 10 combinations where all our digits match. The probability of that is then
10/1000 = 1/100
So, the situation where the first person's last 4 phone number digits matches those of the second person as well as the 4 digits of his/her own pin has a probability of
(1/100).(1/100).(1/100).(1/100) = 1/10000000
just like we had before.

You can always get back to me.

Regards



---------- FOLLOW-UP ----------

QUESTION: Well Ahmed the situation was true.Believe it or not my pin number to my bank card was my girlfriends last four digits(exactly in sequence) of her telephone number when she was a child.And her pin number is my last four digits of my telephone number.(exactly in sequence),when we met 6 years ago.How is that for probability?
Thank you again,
Andrew

Answer
Hi Andrew,
Nice to hear. But from a probability standpoint its not so surprising, even as unlikely as it seems. Admittedly, its highly improbable in a particular case but not in a general one. I mean, it had to happen to a certain two-people combination.
Just to add, suppose you toss a coin ten times in a row, the sequences
HHHHHHHHHH and HHTHTTTHHT are equally likely with probability (0.5)^10 each, but only the first would be considered remarkable. We tend to think of the second as some random ordering of five heads and five
tails, and there is nothing remarkable about getting five heads and five tails (it has a probability of about 1/4). However, the probability to get them in exactly this order is still (0.5)^10, the same as ten heads in a row.
It seems to be a deeply rooted instinct in us humans to look for patterns, regularities, and coincidences and so long as it makes us happy its worth cherishing.

Regards (to both of you).