As Title suggests trying to figure how to solve something in excel specifically.
An event has a $1/3$ Chance to occur. If I do this $78$ Times whats the chance that:
- It happens $0-20$ Times
- It Happens $21-40$ Times
- It Happens $41-78$ Times
By it happens I mean the $1/3$ event.
Consider that the probability of it happening exactly $k$ times is $$\left(\frac13\right)^k\left(\frac23\right)^{78-k}\binom{78}k\\=\binom{78}k\left(\frac13\right)^{78}\times2^{78-k}\text.$$
So, the probability of it occuring $0$ to $20$ times is $$\sum_{k=0}^{20}\binom{78}k\left(\frac13\right)^{78}\times2^{78-k}\\ =\left(\frac13\right)^{78}\sum_{k=0}^{20}\binom{78}k\times2^{78-k}\text.$$
(In Excel, preforming such a summation is simple.)
The rest is similar (simply change the numbers on the top and bottom of the sigma).