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Let's base my question on an example.

Let's say we toss a weighted coin with probability of tails = 0,05. We do it 100 times and we observed exactly 0 tails. What I am trying to estimate is the smallest probability, that on certain confidence level the amount of tails will be greater than currently observed tails (in this case 0, but in general I want all the possibilities.)

I can count the probability of having all not tails as (0,95)^100 ~ 0,006=0,6%.

What I want to count is p, such that P(X>=1) is more likely than P(X=0) (X- number of tails) on a confidence level alfa.

How do I approach this? Thanks in advance.

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    $\begingroup$ See stats.stackexchange.com/search?q=rule+of+3. $\endgroup$
    – whuber ♦
    Commented Aug 18, 2022 at 22:35
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    $\begingroup$ Hi, thanks for response. It fixes the problem if there are 0 tails then, but it doesn't fix the problem when there are some occurrences of tails. $\endgroup$
    – Andrzej Pawelec
    Commented Aug 18, 2022 at 23:02
  • $\begingroup$ As you might imagine, this is a common situation: see our posts on Binomial confidence intervals. $\endgroup$
    – whuber ♦
    Commented Aug 19, 2022 at 14:12

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