Cumulative binomial distribution sum manipulation

Question

I have a binomial distribution, with Random Variable Y and n trials. r is an integer. How can I show that P(Y ≥ r) = P(X ≤ n − r), such that Y is a random variable with probability of success p, and X is a random variable with probability of success (1- p)? I have tried expressing P(Y ≥ r) as: $$\sum_{x=r}^{n} \binom{n}{x} p^x(1-p)^{n-x} = \sum_{x=0}^{n-r} \binom{n}{x+r} p^{x+r}(1-p)^{n-(x+r)}$$ I need to come up with this in the end: $$\sum_{x=0}^{n-r} \binom{n}{x} p^{n-x}(1-p)^{x}$$ but I am unsure of how to proceed next. Could someone give me a hint?

Answer 1

Recall the identity $$\binom nk = \binom n{n-k}. $$ So $$\mathbb P(Y\geqslant r)=\sum_{k=r}^n\binom nk p^k (1-p)^{n-k} = \sum_{k=r}^n\binom n{n-k}p^k(1-p)^{n-k}. $$ Set $j=n-k$, then the above is equal to $$\sum_{j=0}^{n-r} \binom nj (1-p)^j p^{n-j}, $$ which is $\mathbb P(X\leqslant n-r)$.