Board 8 > Another probability question

(+) Yeah! (+)

A random seven digit number is written down. On average, how many unique digits will appear within it? Provide the probabilities for 1-7.


So for this one, Im assuming itll be four, but it might be five. I have no idea what the probabilities are though.

5

I just took a guess

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@turbopuns3 @SeabassDebeste @xp1337 @Paratroopa1 @MartinFF7

8

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@azuarc

@tyder21 @Tom_Bombadil @pyresword @Xuxon @drpez @epyo @dowolf

There are 9,000,000 seven digit numbers, assuming that starting with 0 is not allowed.

7 unique) 9*9P6=544320 of them with all unique digits. (About 6%.)
6 unique) 9C1*7C2*9P5 to pick your double, to place the double, and then select the other digits = 2857680, or 31.7% [This actually excludes having double zeroes, so it's technically higher.]
1 unique) There are a total of 9 of them (.0001%) that have a single digit.

After that it gets much trickier because if you have 4 unique, is that because you have one quadruplicate, or three duplicates, or a triplicate and a duplicate? So those cases require a little more thought than I have time for right now.

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Wrote a program to do this one for me. Thankfully far easier than the previous problem.

1 Unique: 9 (0.0001%)
2 Unique: 5,103 (0.0567%)
3 Unique: 195,048 (2.1672%)
4 Unique: 1,587,600 (17.64%)
5 Unique: 3,810,240 (42.336%)
6 Unique: 2,857,680 (31.752%)
7 Unique: 544,320 (6.048%)

Average: 5.217031 unique digits

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I think it's uncanny that your numbers agree with mine, yet my calculations for 6 unique explicitly ignore numbers like 1234500.

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I was just staring at your solution for 6 unique digits for the same reason. I'm honestly too tired right now to figure out why your solution accidentally fell into the right value.

The program is as naive as can be though, so I don't think it has a flaw just goes one-by-one from 1,000,000 to 9,999,999 and counts unique digits for each.

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I said that, but continued to stare at it anyway.

It's because your solution accidentally introduces a bunch of 0-leading numbers (e.g. 0112345). This perfectly offsets for the double-0's that you weren't contemplating because you can take any of these 0-leading numbers and map them to valid double-0 numbers. 0112345 -> 1002345.

Accidentally very clever.

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Oh, duh. Just because I chose the double first doesn't preclude the number starting with a unique 0.

And also, it's awesome that somehow works out the same. Very nice catch.

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Amazing work, fellas!

I'm really dumb when it comes to figuring this stuff out, but I'm always interested in reading how you work it out etc

Good work guys.

And keep asking these questions Moon

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tyder21 posted...

I'm honestly too tired right now to figure out why your solution accidentally fell into the right value.

I have always been bad at math and most of this is a foreign language to me, but I feel like this perfectly describes my incredibly mediocre high school math career and I laughed out loud when I read it.


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