I What is the probability that the product of the digits is odd?

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The discussion revolves around calculating the probability that the product of the digits of a natural number is odd. Two solutions are proposed: the first suggests that the product is odd only if all digits are odd, leading to a probability that approaches zero as the number of digits increases. The second solution considers the probability of having at least one even digit, concluding that the probability of all digits being odd decreases with more digits, yielding different probabilities for one-digit and multi-digit numbers. Discrepancies arise regarding the treatment of zero and the inclusion of leading zeros in digit counts. Ultimately, both solutions highlight the complexity of determining the probability based on digit structure and distribution.
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What is the probability that the product of the digits in any selected positions of a randomly chosen digit natural number is odd?
I have two solutions, but I have no idea which solution correct. Both of them seem correct to me. Help please.

First solution:
There is an infinite number of natural numbers.
Product of digits is odd only if all digits in number are odd.

Counting natural numbers with all odd digits:

1-digit: 5
2-digit: 5*5
3-digit: 5*5*5
4-digit: 5*5*5*5=5^4

n-digit: 5^n

If we want probability for natural numbers up to n-digits:

Total number of odd-product numbers is :

5+5^2+5^3+…+5^n=5*(1+5+5^2+…+5^(n-1))=5*(5^n-1)/(5–1)=5/4*(5^n-1)

p(n)=5/4*(5^n-1)/(10^n-1)

If n is going to the infinity p(n) is going to 0.

Second solution:
Depends on the length of the digit string.

The more digits there are in a number, the greater the probability that at least one of them is even. If any of them are even, the product is even.

So really the question reduces to the probability that all the digits are odd, and that’s just inverse powers of 2…

1 digit: 1/2 = 50%

2 digits: 1/(2²) = 25%

3 digits: 1/(2³) = 12.5%

and so on.
 
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littlemathquark said:
First solution:
There is an infinite number of natural numbers.
There is no uniform distribution on the natural numbers. It's only possible on a finite subset.

The best you can do is to take the limit as ##N \to \infty## for a subset of ##N## natural numbers. In this case, it's pretty clear that the proportion of natural numbers with all odd digits goes to zero as ##N \to \infty##.
 
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The probability is zero.
 
littlemathquark said:
TL;DR Summary: What is the probability that the product of the digits in any selected positions of a randomly chosen digit natural number is odd?

I have no idea which solution correct. Both of them seem correct to me.
They are both correct.
 
But one of them p(1)=5/9 but for other solution(second) p(1)=1/2.which is correct?
 
littlemathquark said:
But one of them p(1)=5/9 but for other solution(second) p(1)=1/2.which is correct?
Maybe in the first case you didn't count zero as an even number and the second case you did
 
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Hornbein said:
Maybe in the first case you didn't count zero as an even number and the second case you did
And not only ## 0 ##.
In the second solution the original poster has included ## 00 ## into 2-digit natural numbers, ## 000 ## into 3-digit natural numbers, ## 0000 ## into 4-digit natural numbers and so on, and that is wrong.
 
What is the probability that the product of the digits in any k selected positions of a randomly chosen n-digit natural number is odd?
What about this question? Are solution the same?
 
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littlemathquark said:
What is the probability that the product of the digits in any k selected positions of a randomly chosen n-digit natural number is odd?
What about this question? Are solution the same?
That's just slightly tricky in that an n-digit number has a first digit that is randomly 1-9 and n-1 digits that are randomly 0-9.

You could calculate the two cases where the first digit is or is not in the k digits separately.
 
  • #11
The digits that make a number even are 2,4,6,8. So 6 of the digits will keep the product odd.

0.6^(n-1)*5/9 = 0.6^n * 25/27
 
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Hornbein said:
The digits that make a number even are 2,4,6,8. So 6 of the digits will keep the product odd.

0.6^(n-1)*5/9 = 0.6^n * 25/27
On second thought, if any one of the digits is zero then the product is zero. Is that even or odd?
 
  • #13
Hornbein said:
On second thought, if any one of the digits is zero then the product is zero. Is that even or odd?
Zero is even.
 
  • #14
PeroK said:
That's just slightly tricky in that an n-digit number has a first digit that is randomly 1-9 and n-1 digits that are randomly 0-9.

You could calculate the two cases where the first digit is or is not in the k digits separately.
Can you give a solution? I can not solve fully.
 
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