Ten non-independent trials, a 10% chance each (in the prior state of knowledge, not conditional on previous results,), and only one trial can succeed. You satisfy these conditions with something like “I hid a ball in one of ten boxes”, and the chance really is 100% that one is a “success”.
Regardless of whether the trials are independent, the maximum probability that at least one is a success is the sum of the probabilities per trial. In this case that doesn’t yield a useful bound because we already know probabilities are below 100%, but in general it’s useful.
Yeah, it’s cool that “I did n trials, with a 1/n chance each, so the probability of at least one success is… ” does have a general answer, even if it’s not 100%. Just noting that it’s not the only small modification of the title yielding a useful and interesting correct statement.
The ones that came to my mind still involved the sum of the per-trial probabilities. If it was clear that we were looking for something preserving the “n trials with 1/n chance”, rather than the summation, I think it would have been more obvious where you’re going with this.
My guesses at what the spoiler was going to be:
Ten non-independent trials, a 10% chance each (in the prior state of knowledge, not conditional on previous results,), and only one trial can succeed. You satisfy these conditions with something like “I hid a ball in one of ten boxes”, and the chance really is 100% that one is a “success”.
Regardless of whether the trials are independent, the maximum probability that at least one is a success is the sum of the probabilities per trial. In this case that doesn’t yield a useful bound because we already know probabilities are below 100%, but in general it’s useful.
Yeah, it’s cool that “I did n trials, with a 1/n chance each, so the probability of at least one success is… ” does have a general answer, even if it’s not 100%. Just noting that it’s not the only small modification of the title yielding a useful and interesting correct statement.
The ones that came to my mind still involved the sum of the per-trial probabilities. If it was clear that we were looking for something preserving the “n trials with 1/n chance”, rather than the summation, I think it would have been more obvious where you’re going with this.
I thought it would be linearity of expectation.
I appreciate people playing along :)