Are there no mathematicians left in sci.math?

Have abusive deranged flametrolls like the former campus
dishwasher Ludwig Hansen nee Poehlmann or Kessler, 72,
born in Arschlick in the Reich, a lunatic who somehow
eluded the police all these years, driven all the
competent ones away?

The problem recited below is an interesting one, but not
one of those replying seems to have even understood it,
and their comments are either correct but irrelevant (eg
Sergi o's "dont play roulette! Just send the Casino a
check and save on the airplane and hotel, save the time,
go fishing...") or wrong / absurd / nonsensical.

Is sci.math a mathematician free zone?

THE PROBLEM

Introduction/Background

Trivially, when a fair coin is tossed, ad infinitum:

a) Any arbitrarily-long sequence of consecutive heads
will occur, eventually.
So, for example, the probability that a googolplex-long
sequence of consecutive heads will occur is 1.

b) The sequence will occur infinitely often.

a) and b) remain true even if the coin is non-fatally loaded/biased, i.e., for any probability p > 0 of a toss resulting in a head.

Trivial, as I said - which should make the following
problem also trivial:

The Challenge

The coin starts off being fair, i.e. p_1 = 0.5

Then, for each successive toss, the coin is progressively biased against heads by a constant factor B > 1
So:
p_2 = 0.5 / B
p_3 = 0.5 / B^2
..
p_n = 0.5 / B^(n-1)

Let P_B,k = Probability that somewhere within the infinite sequence of toss results there will be at least one occurrence of k > 0 consecutive heads.

QUESTION:
Find the range of values (B,k) for which P_B,k = 1

HINT
P_k = 1 for all B>1, k>0 is not the answer.

I expect such a subject line prompts many who could solve it to reach
for the kill-file button.

Sylvia.

On 10/17/2022 4:49 PM, Perrti Lounesto Jr. wrote:
> Are there no mathematicians left in sci.math?
>
> Have abusive deranged flametrolls like the former campus
> dishwasher Ludwig Hansen nee Poehlmann or Kessler, 72,
> born in Arschlick in the Reich, a lunatic who somehow
> eluded the police all these years, driven all the
> competent ones away?

The letters in
"Archimedes Plutonium"
may be re-arranged to spell
"Penis Could Mature Him"

On 18-Oct-22 12:29 pm, Sylvia Else wrote:
> I expect such a subject line prompts many who could solve it to reach
> for the kill-file button.
>
> Sylvia.

Oops - replying to an old post. Oh well.

Sylvia.

Perrti Lounesto Jr. explained :
> Are there no mathematicians left in sci.math?

You have to know whom to ignore.

> Have abusive deranged flametrolls like the former campus
> dishwasher Ludwig Hansen nee Poehlmann or Kessler, 72,
> born in Arschlick in the Reich, a lunatic who somehow
> eluded the police all these years, driven all the
> competent ones away?

Looks like you are just trolling with this remark.