Probability of finding a 3 digit number six times within the first 1000 digits of pi

Question

What is the probability of six (or more) appearances of a $3$ digit string within the first $1,000$ digits of $\pi$ (assuming $\pi$ was random). Not including rep digits, which may complicate the answer because of their increased likelihood (eg $1111$ could count as $2$ incidences of $111$).

I have calculated that the probability of finding at least one occurrence of a $3$ digit number within a random $1000$ digit string is $0.6323$ based on the formula

$$1-{(1-{0.1}^{3})}^{1000}$$

But the probability of six or more occurrence? Must be a longshot. Any help would be appreciated.

Answer 1

In your 1000 digits you have 998 sequences of 3 digits. The probability to find there exactly $k$ times a given 3-digit number is: $$ p(k)=\binom{998}{k}q^k(1-q)^{998-k}, $$ where $q=1/1000$. The probability of six or more occurrences can be then computed as: $$ 1-p(0)-p(1)-p(2)-p(3)-p(4)-p(5)\approx 0.000582. $$

Answer 2

It is relatively safe to assume the digit triples are independent uniformly random numbers. Also, $1000$ is large enough to assume (esp. given the other approximation assumptions) a Poisson distribution for the the number of occurances of any given triple. Thus out of $1000$ possible triples, we expect a fraction $\frac 1e$ not to occur at all, also $\frac 1e$ to occur exactly once, $\frac1{2!e}$ exactly twice, and so on. Hence we expect about $$ \frac{1000}{e}\left(\frac1{0!}+\frac1{1!}+\frac1{2!}+\frac1{3!}+\frac1{4!}+\frac1{5!}\right)\approx 999.4$$ triples to occur at most five times. Thus a triple occuring at least six times is not too surprising, given that we expect there to be about $0.6$ such triples.

Admittedly "not too surprising" is not really a useful statement about a propability. We can use Poisson again. As the expected number of successes is $\approx 0.6$, the probability of no success is $\approx e^{-0.6}\approx 0.55$, hence the probability of at least one success is $\approx 0.45$, just slightly less than guessing a coin toss tight.