# The Binomial Distribution

- Created by: Danny Hillman
- Created on: 16-05-11 11:20

**The Binomial Distribution**

**The binomial distribution is a distribution which works for any set of events which has:**

**A fixed number of trials**

**Only two possible outcomes from each trial**

**The same probability for each outcome from each trial**

**Trials independent of each other**

**E.g. I flip a coin. It is either heads or tails. It cannot be both and it cannot be anything other than heads or tails. We can also say that I will achieve a result of heads or not heads. **

**To make this into a binomial distribution, we need to know a few things. **

**Number of trials = n**

**Number of successes = r**

**Probability of success = p**

**Probability of failure = q**

**This leads us to the equation:**

**P(X=r) = (nCr)(p^r)(q^(n-r)) where X ~ B(n, p)**

**x ~ B(n, p) simply means X is distributed binomially with number of trials, n, and probability of success, p.**

**If you have a graphical calculator which has a statistics setting, go to that and then find:**

**Dist (Distribution)**

**Binm (Binomial)**

**BcD (which gives you X=r)**

**or**

**BpD (which gives X** ≤ **r)**

**Mean and Variance**

**The mean and variance of the binomial distribution can be found easily using nothing more than n and p.**

**Mean = np**

**Variance = np(1-p)**

**The standard deviation of the binomial distribution can be found by taking the square root of the variance.**

**These two can be found in the formula book on**…

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