Jules César gives Astérix a biased coin which produces heads 70% of the time, and asks him to play one of the following games:
Game A : Toss the biased coin 99 times. If there are more than 49 heads, he will be sent to feed crocodiles.
Game B : Toss the biased coin 100 times. If there are more than 50 heads, he will be sent to feed crocodiles. If there are exactly 50 heads, he is granted a fair coin. If the fair coin produces a head, he will be sent to feed crocodiles. Which game will Astérix choose to play?
My hunch is that it will be game A, but I want to calculate the probability. For Game A, I wanted to compute the probability that he would get more than 49 heads. Using the binomial distribution, I found:
$$\sum_{i=49}^{99}{99 \choose i} (0.70)^i (0.30)^{ 99-i} $$
I tried calculating this through Wolfram Alpha but got 0.999999 which doesn't seem right. Is there an easier way to compute this sum, or probability for that matter?