Understanding Binomial Distribution with Excel 2007

In this post we will try to understand how the Excel built-in Binomial distribution function can be used for calculating Binomial probabilities. Traditionally a binomial random variable is described by the following formula: 
$$b(x,n,p)=\binom{n}{x}p^x q^{n-x}$$

Where $p$ is the probability of success and $q$ is the probability of failure and can be related by the relation $p+q=1$, $n$ is the number of trails and $x$ is the value at which we want to evaluate the binomial probability .

The Excel 2007 has a built-in function of BINOMDIST() that takes arguments of $x,n,p$ and logical arguments for cumulative or point density estimate and return the value of Binomial Variable.

The Syntex of the function is :BINOMDIST(number_s,trials,probability_s,cum) 

Now we will see how the calculation that are done manually can be skipped if you use BINOMDIST() function, lets consider this example:

Find the probability of obtaining exactly three 2's if an ordinary dice is tossed 5 times.

Solution: The probability of success is this case is $p=\frac{1}{6},n=5,x=3$ gives:
$$b(3,5,\frac{1}{6})=\binom{5}{3}(\frac{1}{6})^{3}(\frac{5}{6})^{2}=\frac{5!}{3!2!}\cdot\frac{5!}{6!} = 0.032$$
Using Excel: The same can be calculated by  the formula: 

=Binomdist(3,5,0.1667,False)=0.032

Example # 02: The probabilities that a patient recovers from a rare blood disease is 0.4. if 15 people are known to have contracted this disease, what is the chance that (a) at lease 10 will survive (b) from 3 to 8 will survive and (c) exactly 5 will survive?

Solution: 

(a) For Probability $that at lease 10 will survive$:

$$P(x\geq 10) = 1-P(x\leq 10) = 1 - \sum_{x=0}^{9}b(x,15,0.4) = 1-0.9662 = 0.0338$$
(b) For Probability $that from 3 to 8 will survive$
$$P(3\leq x\leq8) = \sum_{x=3}^{8}b(x,15,0.4) = \sum_{x=0}^{8}b(x,15,0.4)-\sum_{x=2}^{8}b(x,15,0.4)$$
$$=0.9050-0.0271 = 0.8779$$
(c) For Probability $that exactly 5 will survive$
$$P(x = 5) = b(x,15,0.4) = \sum_{x=0}^{8}b(x,15,0.4)-\sum_{x=0}^{4}b(x,15,0.4)$$
$$=0.4032-0.02173 = 0.1859$$
Using Excel:

(a) =1-Binomdist(9,15,0.4,TRUE) = 0.0338
(b) =Binomdist(8,15,0.4,TRUE)-Binomdist(2,15,0.4,TRUE) = 0.8779
(c) =Binomdist(5,15,0.4,TRUE)-Binomdist(4,15,0.4,TRUE) = 0.1859

hence we can see how easily we can estimate the binomial probabilities using Excel 2007, I hope that you will like this post. Please give your feedback to improve the blog.