Odometer Sense

So I have been learning to drive, and every time I look at my odometer it gives me two five digit numbers for my mileage that are always different yet somehow almost always have the same digits. This occurs at what seems to be a disproportionately high frequency. So I was wondering what is the probability that given two five digit numbers that they share the same digits?

E.g. 45665 and 44566; 12345 and 54321; or 11112 and 22221; but not 23775 and 23765.

#Combinatorics

Note by Kevin Gao
3 years, 4 months ago

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Comments

It feels do-able if numbers beginning with a zero are allowed, e.g. 02,383, but not sure if not.

Stephen Mellor - 3 years, 4 months ago

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(This includes the numbers beginning with 0). First of all, think of the ways in which you can partition the number into like digits, where each letter represents a different digit (ignore the order for now):

  • AAAAA
  • AAAAB
  • AAABB
  • AAABC
  • AABBC
  • AABCD
  • ABCDE

Now for each of these, calculate the probability of them occurring. For example, each probability is out of 100,000100,000 as there are that many possible 5 digit numbers. For AAAAA, there is 10 possibilities for the first digit and then 1 possibility for each of the remaining digits. This means that the probability of getting an AAAAA is 110000\frac{1}{10000}. Once the probability for each is calculated, then think about the number of ways out of 100000 there are for the second number to match the first number. For example, for AAAAA, there is only 1 number that matches it meaning the probability is 1100000\frac{1}{100000}. If you multiply the probabilities, you will get 1109\frac{1}{10^9} for AAAAA. Calculate each of these multiplications, and then add all of the results together to get the answer.

Stephen Mellor - 3 years, 4 months ago

We have an upper bound of 1/750 since there are 90,000 numbers and each number can be arranged into at most 120 combinations. So collisions are not actually that common -- although if you look at your odometer a couple times a day you're bound to run into one here and there.

Dmitriy Khripkov - 3 years, 3 months ago
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