To the 9s

Probability Level 3

Let $X$ be a random positive integer chosen between 1 and $10^N$ inclusive for some integer $N$ . And denote $P_N$ as the probability that $X$ contains the digit 9.

Evaluate $\displaystyle \lim_{N\to\infty} P_N$ .

\frac{1}{2}

\frac{9}{10}

\frac{1}{9}

1 solution

Romain Bouchard
Jan 25, 2018

Relevant wiki: Paradoxes in Probability

The probability that a random number between $1$ and $10^n$ contains the digit $9$ is $1-(\frac{9}{10})^n$ : it increases as $n$ (and thus the range) increases.

Thus the probability for any positive integer to contain the digit $9$ is then $0.\bar{9} = 1$ .

Though it seems counter-intuitive (or even wrong because there surely are integers who don't contain the digit $9$ ), we need to take into account the fact that we are talking about the probability of an event from a infinite sample space. It means that the event happens almost surely rather than every time (like for a finite sample space.)

@Romain Bouchard We can also see that the convergence of the Kempner Series is roughly based on this fact.......!!!!!

Aaghaz Mahajan - 2 years, 11 months ago

Indeed that one is really mindblowing as well. I did a small problem on it

Romain Bouchard - 2 years, 11 months ago

IKR!!!! I tried your problem as well Sir!!! This is why I like Calculus too much.............mixes all possible fields of Mathematics to bring out the most amazing results possible......!!!

Aaghaz Mahajan - 2 years, 11 months ago

To the 9s

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