Liars poker probability question

A dollar bill has an 8 digit serial number. Assume that each digit in the serial number is randomly generated. What is the is the probability of having 5 of the same digit in the serial number? Scroll down for the correct formula to use. n=8, p=1/9, then slot in k=5,6,7,8 and add them up. It’s going to be a pretty low probability I’d say. Considerably less than one percent.


hmmm - I got 10/100,000,000 or 1/10,000,000 if it were inclusive it would be 40/100,000,000 or 4/10,000,000

I’m no mathematician, nor do I play one on TV, but the probabilities given here seem very low to me. I have 0’s on some of my bills, so that means there are 100,000,000 possible number combinations. Assuming the govt permits all possible combinations, including 00000000, there are 10 occurences where all 8 are the same. There has to be thousands, if not tens of thousands, of occurences of one digit occuring 5 times.

Carson, thanks for the formula. Just a clarification, p should be 1/10, right? (0-9=10 possible digits). Also, I have no real need to know the answer, just thinking about it this am.

Let me know if this is right. 10C1 * 8C5 * 9P3 / 10^8 You can calculate it if you’d like but it’s around a quarter of a percent Edit: whoops forgot to divide by total possible outcomes (10^8)

According to this online calculator, using p=.1, 8 trials, 5 successes: The probability of getting exactly 5 is .00040824 And getting 5 or more is .00043165 So +/- 4 in 10,000 or 1 in 2,500

Yeah you’re right. Probability is 1/10. Doh!

WRONG: 10C1 * 8C5 * 9P3 / 10^8 RIGHT: 10C1 * 8C5 * 9^3 / 10^8 = 0.0482% Wrongly used permutations. Forgot this was with replacement.