Things I Didn't Know I Didn't Know

[jpmassar]

4159 is the first four-digit prime to occur in the decimal expansion of pi.

3 is obviously the first 1-digit prime.
and 31 is the first 2-digit prime.

Other than that, I remain clueless.

Comments

[jpmassar]

Zounds. The first ten digits contain the first 1-6 digit primes. Who'd of thunk?

[jpmassar]

We want the probability that given a random 10 digit number, it will contain EACH of (as consecutive left to right digits)

-- a 1 digit prime
-- a 2 digit prime
...
-- a 6 digit prime

The probability that a random 6-digit number is prime is approximately 8.7%. The probability that at least one of 4 6-digit numbers is prime is thus about 31%.

For 5 5-digit numbers a similar calculation gives about 72%.

So without continuing I'd guess we'd be looking at something like a 15% chance, ignoring issues of independence, which doesn't warrant a 'zounds'.

Once again, probability triumphs over intuition.