recursive formula

[enzocreti]

I dont remember but there is a recursive formula for calculating the chance that six nines in a row occur this early in pi grecus (the Feynman Point so called)...can somebody explain to me how this probability is calculated?

[CRGreathouse]

If you treat each sequence of six digits like six numbers selected uniformly at random from 0-9, the chance that a given one is 999999 is 1/1000000, the chance it isn't is 999999/1000000, and the chance that none of the first 762 are is (999999/1000000)^762 = 0.9992.... But if there's nothing special about the 9, you might look at the chance that none of the first 762 starts a sequence of six digits all alike -- I trust you can find these odds?

[enzocreti]

Originally Posted by CRGreathouse

If you treat each sequence of six digits like six numbers selected uniformly at random from 0-9, the chance that a given one is 999999 is 1/1000000, the chance it isn't is 999999/1000000, and the chance that none of the first 762 are is (999999/1000000)^762 = 0.9992.... But if there's nothing special about the 9, you might look at the chance that none of the first 762 starts a sequence of six digits all alike -- I trust you can find these odds?

There is a recursive formula


the number can end in x (x different from 9)
can ends in x9, x99, x999, x9999,x99999


so the probability not to see six nines in a row in the first 762 digits let's call it 9T(762-1)


9T(761)=9T(760)+9T(759)+9T(758)+9T(757)+9T(756)


this should be the recursive formula but i am not sure


the problem is also to determine the values of T(761) ...

[CRGreathouse]

There is a very simple formula you can punch into a scientific calculator (you need an exponent key, not just a 4-function calculator, but nothing fancy). Look at my post.