# Solved – Relation between binomial and negative binomial I was reading on negative binomial from a Statistics textbook and came across this portion on probability relation between binomial and negative binomial. $$Y$$ refers to the number of trials required to get $$r$$ successes.
Can somebody please explain the relation

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Based on binomial distribution, event $${X geq r}$$ is the set of outcomes that satisfy "$$n$$ trials led to $$r$$ successes or more", which is equivalent to "$$r$$-th success happened at $$n$$-th trial or before", which is in turn equivalent to "$$n$$ trials or less were required to get $$r$$ successes", and that is it. begin{align*} P{X geq r} &= P{mbox{at least r successes in n trials}}\ &= P{mbox{r-th success in n-th trial or before}}\ &= P{mbox{n or fewer trials to get r successes}}\ &= P{Y leq n} end{align*}
The second relation is the complement of first relation that is: begin{align*} P{X geq r} &= P{Y leq n},\ 1 – P{X geq r} &= 1 – P{Y leq n},\ P{X < r} &= P{Y > n}\ end{align*}
$$P{mbox{less than r successes in n trials}}= P{mbox{more than n trials to get r successes}}$$