Maximizing Seed Germination: Calculating Probability without Binomial Theorem

[eldrick]

A certain type of seed has a probability of 0.8 of germinating. In a pack of 100 seeds, what is the probability that at least 75% will germinate?




Solution can be achieved on a calculator using binomial theorem. Is there any other way of doing it without using binomial theorem ?

 

[HallsofIvy]

You could use the normal distribution approximation, with "half- integer correction".

A binomial distribution in which the probability of a single success is p, repeated n times, has mean $\mu= np$ and standard deviation $\sigma= \sqrt{np(1-p)}$. If n is large, the normal distribution with the same mean and standard deviation is a good approximation.

Since a normal distribution allows real number values while a binomial distribution requires integers, you interpret any real number that rounds to a particular integer as being that integer. Here, "at least 75%" or "at least 75 out of 100" would be equivalent to "74.5 or larger". That's the "half integer correction".