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I am performing a product trial. The trial has three possible outcomes with the following probabilities:

Outcome A = .3  Outcome B = .5  Outcome C = .2 

If I perform five trials, what are the odds of B and C occurring at least once throughout testing?

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The complement of this event is that only $A$ and $C$ occur or only $A$ and $B$ occur. The probabilities for those are $(.3+.2)^5=.5^5$ and $(.3+.5)^5=.8^5$, respectively. However, adding these two double-counts the possibility that only $A$ occurs, so we have to subtract $.3^5$, for a total of

$ .5^5+.8^5-.3^5=0.3565\;. $

Thus the probability of your event is $1-0.3565=0.6435$.

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    To add to my previous comment, you say that the complementary outcomes are only A, C or only A,B. What about only A occurring?2012-09-27
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This is the complement of the probability that all five are outcome A. Assuming independence this complement is 1 - .3^5 = .99757

Sorry this is for B or C occurring at least once. Please downvote more.

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    @Imray: $Y$es, that's right. Sorry I didn't get around to responding earlier myself.2012-09-28