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I have never been able to wrap my head around probability, and I often find that my intuition is wrong. In this case, I don't even have intuition as to where to begin.

If I have two trials, each with a 1/16 chance of success, what are the chances that either or both of them result in success? How, mathematically, do you arrive at the correct probability? How, intuitively, can I understand this number?

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The event you're interested in is the complement of the event that both trials fail. The chance of one trial failing is $1-1/16=15/16$, so the chance of both trials failing is $(15/16)*(15/16)=225/256$. So the chance of the opposite happening, which is what you're interested in, is $1-225/256=31/256$. – joriki Apr 6 '11 at 13:32
To help your intuition: what is the probability of a failure if you have one trial? if you have two trials? – Did Apr 6 '11 at 13:33
@joriki Why is that a comment and not an answer? – StrixVaria Apr 6 '11 at 13:33
@StrixVaria: Good question. It seems to be a wide-spread practice here to give short, relatively easy answers in comments and not as an answer, and I've adopted that. My guess is that the idea is that reputation (which you only get for upvotes for answers, not comments) should reflect more difficult answers. Especially since relatively easy answers often get a lot more upvotes than difficult ones because a lot more people understand them. This is just me guessing, though -- perhaps I should have posted an answer instead. – joriki Apr 6 '11 at 13:40
@joriki: A few opinions: Personally I prefer that even simple answers (like for this question) are given as answers instead of comments. For example, a question might have "0 answers" according to the front page, but when one looks at it, there turns out to be a perfectly good answer in a comment already. Annoying. And how many times haven't we seen "please write that comment as an answer so that I can accept it"? Moreover, I think that votes should go to the most useful answers, not necessarily to the most difficult ones. ;) – Hans Lundmark Apr 6 '11 at 14:48
up vote 4 down vote accepted

The probability that neither trial is successful is $(15/16)^2$ (assuming that the trials are independent), and the chances that at least one trial is successful is one minus that: $1-(15/16)^2 = 31/256$.

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