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Among the positive integers less than 10 , 000 , 000 10,000,000 , do the majority contain the digit 1?

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3 solutions

Nicholas James
Mar 4, 2017

As is often the case, it's easier to count the integers below 10000000 10000000 which do not contain the digit 1 1 .

This leaves 9 9 possibilities for each digit: 0 0 and 2 2 to 9 9 .

So, for the integers up to 9999999 9999999 , we have 9 7 1 = 4782968 9^7-1=4782968 integers which do not contain the digit 1 1 . We had to subtract 1 1 because we had counted 0 0 as one of our possibilities.

Therefore, 9999999 4782968 = 5217031 9999999-4782968=5217031 integers contain the digit 1 1 , which is the majority.

Colin McMullen
May 10, 2017

I cheated & wrote a programme, 52.2 % to 1 dec. place.

Wuu Yyiizzhhoouu
May 9, 2017

the probability of any digit not being a 1 is 0.9. the probability of numbers with 7 digits or less not having 1 in them is 0.9^7=0.478(3sf). Thus the probability of such numbers containing 1 is (1-0.478)>0.5, thus the majority contain 1.

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