Binomial Distribution

October 4, 2020

In the last section, we took a first look at the some of the conditions necessary for a binomial experiment. Let's examine the binomial distribution more closely by examining the method to calculate the probability of getting a particular outcome for a series of bernoulli trials.

As an example, let our binomial experiment consist of flipping a coin ten times. Suppose for now, we believe the coin is fair, and we are interested in the probability of getting 5 heads. $n$ = 10. Since we are interested in the number of heads, "success" in a trial is the coin landing on heads. $p$= 0.5 since we believe the coin is fair.

The formula for calculuating the probability of $x$ success in $n$ trials of a binomial experiment is given by $$\text{P}(X=x)=(\genfrac{}{}{0ex}{}{n}{x})p^x(1-p{)}^{n-x}\text{where}(\genfrac{}{}{0ex}{}{n}{x})=\frac{n!}{x!(n-x)!}.$$ The large parenthesis is called a combination. Substituting the numbers stated above into our equations we get the probability that our random variable $X$ takes on a value of 5 is about 0.25. So there is about a one in four chance we get five heads.