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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 $1-225/256=31/256$.2011-04-06
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    To help your intuition: what is the probability of a failure if you have one trial? if you have two trials?2011-04-06
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    @joriki Why is that a comment and not an answer?2011-04-06
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    @joriki Hmmm... *Les grands esprits se rencontrent*, as the saying goes.2011-04-06
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    $joriki: Ha, you beat me be 42 seconds. There should be a "new comments have been posted" notification (just like for answers)...2011-04-06
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    @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.2011-04-06
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    @Hans: Yes -- although I often find that the "new answers have been posted" notification takes a lot longer than 42 seconds to kick in.2011-04-06
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    @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. ;)2011-04-06
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    @Hans: That makes sense to me, but apparently there are different opinions on this (as evidenced by the +1 for both your comment and mine :-). I think I'll look for a discussion on this on meta and start one if I don't find one.2011-04-06
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    @joriki: Here's a meta thread: http://meta.math.stackexchange.com/questions/1559/dealing-with-answers-in-comments2011-04-07
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    @Hans: Thanks very much -- that's quite informative. So I'll post answers as answers in the future.2011-04-07

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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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It also might be helpful to think of this in terms of geometric distribution.

Geometric distribution represents the number of failures before you get a success and is defined as:

$(1-p)^{n-1} \times p$

In your case, what we have is two geometric distributions added together:

$\left(\frac{15}{16}^{0} \times \frac{1}{16}\right)+\left(\frac{15}{16}^{1} \times \frac{1}{16}\right) = \frac{31}{256}$

The reason we add it together is the first term assumes that you had no failure and your first trial was a success, your second term assumes that you had one failure and your second trial was a success.