Consider a sequence of n independent Bernoulli trials with the likelihood of a success p on each individual trial stays constant from trial to trial with \(0 \lt p \lt 1 \text{.}\) If we let the variable \(X\) measure the number of successes obtained when doing a fixed number of trials n with \(R = \{ 0, 1, ..., n \}\text{,}\) then the resulting distribution of probabilities is called a Binomial Distribution.
Now, let’s determine the actual probability function for this distribution.
Let’s consider a simple example for flipping coins. Indeed, suppose you flip a coin exactly 20 times and need to determine the probability of getting exactly 10 heads.
This is binomial with n = 20, p = 1/2 and you are looking for f(10). With these values
Note that the mean for this distribution is also 10 so one might expect 10 heads in general
If you rather would prefer to determine the probability of getting 10 or fewer heads requires F(10) = f(0) + f(1) + ... + f(10). There is no "nice" formula for F but this calculation can be performed using a graphing calculator, such as the TI-84 with F(x) = binomcdf(n,p,x). In this case, F(10) = binomcdf(20,1/2,10) = 0.588.
Often one will test a critical system components for failure and toward that end collect a sample of 100 of these components from the manufacturer. Suppose the component is listed as having a p = 0.01 probability of breaking and you want to know the likelihood that at most 1 of the tested components actually fails when tested. You find it reasonable to presume that different components succeed or fail independently of each other. So, you can model this situation with a binomial distribution.
If \(X\) measures the number of components that fail when tested, the specific probability function is given by
The skewness 3 and kurtosis 4 can be found similarly using formulas involving E[X(X-1)(X-2)] and E[X(X-1)(X-2)(X-3)]. The complete determination is performed using Sage below.
The following uses Sage to symbolically confirm the general formulas for the Binomial distribution.
The Census Bureau reports that 82% of Americans over the age of 25 are high school graduates. A survey of randomly selected residents of certain county included 1260 who were over the age of 25, and 1086 of them were high school graduates.
(a) \(\) Find the mean and standard deviation for the number of high school graduates in groups of 1260 Americans over the age of 25.
Mean =
Standard deviation =
(b) \(\) Is that county result of 1086 unusually high, or low, or neither?