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A die is loaded to give the probabilities:

$$P(1)=0.3$$ $$P(2)=0.1$$ $$P(3)=0.15$$ $$P(4)=0.15$$ $$P(5)=0.15$$ $$P(6)=0.15$$

and thrown 8 times. Find the probability that both 1 and 2 occur.

This problem seems funnily easy but I can't see how to solve it without brute-forcing all the combinations...

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up vote 3 down vote accepted

Hint: You can use inclusion/exclusion. Calculate the chance that 1 doesn't occur, which is $0.7^8$ and the chance that 2 doesn't occur. Add them together, but you have double counted the cases where neither occurs, so deduct the chance which neither occurs.

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It is easier to find the probability that $1$ or $2$ or both fail to occur.

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Ah, misread "both fail." – Thomas Andrews Apr 18 '13 at 20:49

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