# 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

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*}
$$P\{\mbox{less than r successes in n trials}\}= P\{\mbox{more than n trials to get r successes}\}$$
• In the second relation, is it right to say that the P( less than r successes in n trials) $=$ P( more than n trials to get r successes) Mar 16, 2019 at 10:15