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Say that I have an event that happens with probability $p$ -- lets say the probability of having a $6$ when I throw a die. Is there a formula that tells me the probability of have at least one of these events if I have $n$ simultaneous trials? Let's say I throw $4$ dice, what is the probability of having at least one $6$?

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We can use something called complementary counting to find the probability that an event doesn't happen. Then we subtract this probability from $1$ to find the probability that the event does happen. In your example, if you have $n$ trials and the probability of getting a 6 is $p$ on each trial, then the probability that any given roll is not a 6 is $1-p$. Then out of $n$ trials, the probability that none of them are a 6 is $(1-p)^n$. Then we see that if we take $1-(1-p)^n$, this gives us the probability that at least one of the trials is a 6.

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