This is probably (...) a very basic probability question, but I'd like to be sure about it.

I have 6-sided dice. When rolled, I count one success per dice if the number on the dice is 5 or more.

for each dice, I have a probability of 1/3, but what's the probability of having at least 1 success when rolling n dice at a time?

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    $\begingroup$ This is a generalization of the question asked at stats.stackexchange.com/questions/53154, which is directed at the cases $n=1$ and $n=2$. One of the answers gives a formula for all $n$ and any types of dice (not just six-sided): see "Solution 2" at stats.stackexchange.com/a/53160. $\endgroup$
    – whuber
    Commented Jan 15, 2015 at 20:19
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    $\begingroup$ The usual trick on "at least one success" type problems is to work out the probability of the complementary event ("no successes"), which is usually simple enough to write down immediately. $\endgroup$
    – Glen_b
    Commented Jan 16, 2015 at 1:21

1 Answer 1


You are describing the binomial distribution. The binomial distribution specifies the probabilities of $x$ number of successes in $n$ independent trials each with probability of success $p$. In your case, $x=1$ and $p=1/3$.

You can answer this question with the following calculator: http://stattrek.com/online-calculator/binomial.aspx. You can also do this in R with the function pbinom.

If you are interested in learning more about the binomial distribution and want a good textbook I recommend A First Course in Probability by Sheldon Ross. If you want a less in-depth look, I recommend this video: https://www.khanacademy.org/math/probability/random-variables-topic/binomial_distribution/v/binomial-distribution

  • $\begingroup$ Great, thanks! would you have an idea of the formula (for P(X >= 1)) used on the website? I need to program it from scratch. $\endgroup$
    – Lucien S.
    Commented Jan 15, 2015 at 18:51
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    $\begingroup$ $P(X >= 1) = 1 - P(X=0) = 1 - (1-p)^n$; I highly recommend the Khan Academy video. $\endgroup$ Commented Jan 15, 2015 at 19:37

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