How to know the variability of Expected Value for a sample series of events with different outcome probabilities?

I have a quick question I would humbly like to ask for your help to solve: let's assume that I am analyzing a sample series of events with different probabilities of success. There are only 2 possible results: "success" or "failure".

See the table below for reference: To find the EV of this set, I took the number of Successes (6) and from that, subtracted the sum of all percentages in the column a_prob (7,41). Therefore, 6 - 7,41 = -1,41, which should be the EV for this set.

What I would like to know now is by how much this EV can vary, considering that this data set is a sample of a bigger universe. My first idea was to calculate the Standard Deviation, but the EV is a constant, so I am not sure if I can apply StDev to that.

So first question would be: how can I know the variability of the number?

And second: how can I calculate that using Excel?

Obs: My knowledge in statistics is very limited. I am still learning - please let me know if my question is formulated poorly or if it is just plainly stupid haha.

Thank you so much in advance!

EDIT 1: Grammar, correcting mistakes. And thanks for the answers so far - the question was worded poorly before. I hope it is understandable now.

• You say "The results can be either "success" or "failure", so the distribution is binomial." but also say the sucess probabilities are unequal. But then the distribution is not binoial, which assumes constant success probability. – kjetil b halvorsen Jul 30 '15 at 19:27
• Thank you for your comment! As you can see, I lack knowledge on many basic statistics concepts - I thought that what made the distribution binomial was the idea that only 2 results were possible, and not necessarily the probability of each result happening. Thank you for helping me understand this =)! – sharpbounce Jul 30 '15 at 20:00
• You seem to be using "expected value" to be something like "sample mean". It isn't. The expected value is a property of a probability distribution; if the situation is unaltered so that the underlying distribution is unchanged, expected value doesn't change.. – Glen_b Aug 1 '15 at 10:34
• Thanks Glen_b. This is very helpful knowledge. From what I understand, what I am trying to calculate is not needed then, correct? Expected Value should not have any variability as long as the probabilities taken into account to calculate it don't change. Please, let me know if this thought process is not correct. – sharpbounce Aug 1 '15 at 18:28

First of all, "standard deviation of expected value" doesn't make sense. The expected value is a constant and its standard deviation is zero. Do you mean the s.d. of the sample proportion? If so this can be found in a straightforward way. Letting $X_i$ be an indicator equal to one if trial $i$ is a success and zero otherwise and assuming independence across trials \begin{align} \text{Var} \left ( \frac{\sum_{i=1}^{n} X_i}{n} \right ) &= \frac{1}{n^2} \sum_{i=1}^{n} \text{Var}(X_i) \\ &= \frac{1}{n^2} \sum_{i=1}^{n} p_i(1 - p_i) \end{align} and so the standard deviation is $n^{-1} \sqrt{\sum_{i=1}^{n}p_i (1 - p_i)}$.