What is a cumulative Binomial probability? I am new to this so don't know if I am asking the right question as I just read about its usage but didn't know what exactly a cumulative Binomial probability is.
So my question is: What are cumulative Binomial probabilities? Any example will be of great help.
 A: TL;DR - Skip down to the ===== section to read just the example:
I'll start off with a random variable that is not binomial, but will provide an easy to understand example: a uniform random variable. An example of an event that has a uniform distribution is rolling a dice; each of the outcomes is equally likely to occur (1/6 probability for a 6 sided die). So if I asked, "what is the probability that a 1 is rolled?" You would say 1/6. In notation, if $X$ is a (discrete) random variable with a uniform distribution over the integers {1,2,3,4,5,6} then $$\text{Pr}\left\{X = 1\right\} =\cdots=\text{Pr}\left\{X = 6\right\}= 1/6$$
One often writes this as a function in terms of $x$, i.e. $p(x) = \text{Pr}\left\{X = x\right\}$. This is often called the probability mass function or pmf (for continuous random variable there is an analog called the probability density function or pdf). 
The cumulative mass/density function (or sometimes called distribution function) is $$F(x) = \text{Pr}\left\{X \leq x\right\}$$ So in our uniform example it is what is the probability of rolling an $x$ or lower (e.g. What is probability of rolling a 3 or less?).
A binomial random variable, $X\sim \text{Binomial}(n,p)$, is characterized by two parameters, $n$ and $p$, then number of trials and the success probability at each trial. See wikipedia for more information on definition of a binomial. The binomial has pmf $$\text{Pr}\left\{X = x\right\} = {n \choose x}p^x(1-p)^{n-x}$$ and distribution function $$F(x) = \text{Pr}\left\{X \leq x\right\}=\sum_{k=1}^x {n\choose k}p^k(1-p)^{n-k}$$ So that was a lot of math how about some examples:
A binomial is characterized by a random phenomenon in which there are (1) $n$ independent trials, each (2) dichotomous (i.e. success/failure, yes/no, 1/0), and (3) probability of success at each trial is constant, $p$. 
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A classic example is flipping a coin $n$ times. Lets say we flip a fair coin 10 times and we are interested in the number of heads. Then the random variable $X = \text{Number of heads out of 10 flips}$ is $X\sim\text{Binomial}(n=10,p=1/2)$. We could ask questions such as, what is the probability we get exactly 5 heads? We could answer this using the pmf: $$\text{Pr}\left\{X = 5\right\} = {10\choose 5}(1/2)^5(1-1/2)^{10-5}=252*(0.03125)*(0.03125) = 0.2460938$$ or we can ask a question like: "What is the probability of getting 5 or less heads?" This is a cumulative binomial probability. We use the distribution function to get an answer: 
$\begin{align*}
\text{Pr}\left\{X \leq 5\right\} &= \sum_{k=1}^5 {10\choose k}(1/2)^k(1-1/2)^{10-k}\\
&= (0.5)(0.0009765625) + 10*(0.5)(0.001953125) + 45(0.25)(0.00390625) + 120(0.125)(0.0078125) + 210(0.0625)(0.015625) + 252(0.03125)(0.03125)\\
&= 0.6230469
\end{align*}$
it is cumulative in the sense that it 'accumulates' probability, i.e. probability of getting 5 or less heads is equal to probability of getting 0 heads PLUS probability of getting 1 head PLUS probability of getting 2 heads ...
A: I thought k=0 then B(x;n,p) 
If k=1 then it should be B(x-1;n,p)
where B( ) cumulative dists.
There is an identity
b(x;n,p)=B(x;n,p)-B(x-1:n,p)
