I'm studying in an introductory statistics textbook and it's confusing me when it mentioned this:
"Any experiment that has characteristics two and three and where n= 1 is called a Bernoulli Trial(named after Jacob Bernoulli who, in the late 1600s studied them extensively). A binomial experiment takes place when the number of successes is counted in one or more Bernoulli Trials."
You flip a coin. X is the random variable "The outcome of the coin flip", which can be H or T. X is a Bernoulli random variable which, if the coin is fair, has P(X = H) = 0.5 and P(X = T) = 0.5.
Now consider flipping the coin 10 times. Let us say that Y is the random variable "The number of heads out of ten coin flips". Now Y can take any of the values 0 to 10 inclusive. It is a Binomial random variable and quite different from a Bernoulli variable. It then makes sense to calculate the 11 probabilities P(Y = 0), P(Y = 1),...,P(Y = 10).
When considering probability questions, and before you start to calculate anything, you should always try and write down exactly what your variable is e.g. X = "The outcome when I flip a coin once", Y = "The number of heads out of 10" etc.