First, let's review the binomial distribution itself:
Using that formula, you can create a probability distribution for all the values of k: zero successes, one success, two successes, all the way up to n successes.
k  0  1  2  3  ...  n 

P(X = k) 






That can be made into a histogram, where the xaxis represents the values of k, the number of successes; and the yaxis is the relative frequency of those successes. Each bucket for the values of k go up to the height corresponding to the probability.
Just like all distributions, that histogram is going to have a mean and a standard deviation. The mean is fairly obvious to calculate.
EXAMPLE
Suppose you rolled a die six times. How many threes would you expect? What if you rolled it 60 times or 600 times? How many threes would you expect?For a binomial distribution, we can identify the mean, standard deviation, and variance:
Every distribution has three key features:
You just dealt with center and spread by finding the mean and the standard deviation.
But what about the shape? Shape of this distribution is affected by two things: both n and p.
Look at the following distributions for a high, low, and middle probability. Notice how the distribution changes as the probability changes and number of trials change.
High Probability p = 0.90  

Number of Trials  Distribution 
10  
100 
When the probability of success is very high, the distribution is skewed very heavily to the left when the number of trials is low. But as the number of trials increases, the distribution is nearly symmetric. It's a little skewed to the left, but not heavily skewed as with the lower number of trials.
Low Probability p = 0.15  

Number of Trials  Distribution 
10  
100 
When the probability of success is very low, the distribution is skewed very heavily to the right when the number of trials is low. But as the number of trials increases, the distribution is now only slightly skewed to the right.
High Probability p = 0.50  

Number of Trials  Distribution 
10  
100 
When the probability of success is in the middle, near 0.50, the distribution becomes nearly symmetric when the number of trials is low. As the number of trials increases, the distribution stays symmetric.
That's what you should see when we look at the normal distribution approximation of the binomial distribution.
This is a critical concept. This means that when you have a large number of trials, the distribution of binomial probabilities is nearly normal, with the mean of what you found the mean to be, and standard deviation of what you found the standard deviation to be. Ultimately what you're finding is the binomial distribution with parameters n and pwhich is what makes the binomial look like what it looks likelooks a lot like the normal distribution with that mean and that standard deviation.
The distribution has to be large enough to satisfy these two conditions:
This means that you had to be far enough off of the lefthand side and far enough off the righthand side. When you had that distribution, it looked normal when you were safely in the middle of the distribution, and not near the very ends. These two conditions have to be satisfied. This makes looking at a lot of these problems much easier.
EXAMPLE
Suppose a baseball player gets a hit 28% of the time when he comes to bat. What’s the probability that he gets over 30 hits in his next 95 at bats?Source: Adapted from Sophia tutorial by Jonathan Osters.