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I want to know the relationship between binomial and geometic distribution. I know the distribution both have two outcome and probability of success is the same for both distribution.

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Binomial distribution describes the number of successes $k$ achieved in $n$ trials, where probability of success is $p$. Negative binomial distribution describes the number of successes $k$ until observing $r$ failures (so any number of trials greater then $r$ is possible), where probability of success is $p$. Geometric distribution is a special case of negative binomial distribution, where the experiment is stopped at first failure ($r=1$). So while it is not exactly related to binomial distribution, it is related to negative binomial distribution.

If you are looking to learn more about the probability distributions you can check the Statistics 110: Probability lectures by Joe Blitzstein from Harvard University that are freely available online.

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The common definition of the Geometric distribution is the number of trials until the first success (and that's when the experiment stops). See the Wikipedia article https://en.wikipedia.org/wiki/Geometric_distribution . The following is an example for the difference between the Binomial and Geometric distributions: If a family decides to have 5 children, then the number of girls (successes) in the family has a binomial distribution. If the family decides to have children until they have the first girl and then stop, the the number of children in the family has a Geometric distribution (the number can be 1,2,... and is in theory unbounded). A variation on the Geometric distribution is counting the number of failures until the first success, and then the number can be 0,1,2,.... In the example it would be counting the number of boys in the family before the first girl was born, and not the total number of children. The difference between the two variations is always 1.

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