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I've got a binomial distribution where I'm trying to figure out the probability of success (specifically, the odds of a certain event in a video game). This would be simple enough, except that I don't know for certain how many trials I've done. I've got an upper bound (roughly 90,000), and I know the exact number of successes (58), but up to 10% of the trials may not have been recorded (with no bias in which ones were lost). How can I handle this?

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    $\begingroup$ Does your count of successes include the lost trials? Without strong assumptions, you're not going to be able to do much more than give bounds on the parameter. $\endgroup$
    – Glen_b
    Sep 26, 2014 at 8:49
  • $\begingroup$ @Glen_b, A lost trial is completely lost, recorded as neither a success nor a failure. I know how many trials there should have been, and I know how many of the trials that actually took place were successes, but I've only got an estimate for how many trials actually took place. $\endgroup$
    – Mark
    Sep 26, 2014 at 9:40
  • $\begingroup$ Okay, I think I get it. You just don't know the actual number of trials that generated the count of successes. You'll be able to give upper and lower bounds on the observed proportion. $\endgroup$
    – Glen_b
    Sep 26, 2014 at 9:46

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All depends on the nature of the missingness process. If the reason for missing an observation has nothing to do with the binomial $p$, neither with the value success/failure itself, the process is Missing at Random. Then there is no information, and you can analyze as there was no missing at all. Otherwise, we would need more information about context and missingness process.

With no such information at all, the best you can do is give upper/lower bounds on observed proportion.

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