Example 101 Section
We previously looked at an example in which three fans were randomly selected at a football game in which Penn State is playing Notre Dame. Each fan was identified as either a Penn State fan (\(P\)) or a Notre Dame fan (\(N\)), yielding the following sample space:
\(S = \{PPP, PPN, PNP, NPP, NNP, NPN, PNN, NNN\}\)
We let \(X\) = the number of Penn State fans selected. The possible values of \(X\) were, therefore, either 0, 1, 2, or 3. Now, we could find probabilities of individual events, \(P(PPP)\) or \(P(PPN)\), for example. Alternatively, we could find \(P(X = x)\), the probability that \(X\) takes on a particular value \(x\). Let's do that (again)! This time though we will be less interested in obtaining the actual probabilities as we will be in looking for a pattern in our calculations so that we can derive a formula for calculating similar probabilities.
Solution
Since the game is a home game, let's again suppose that 80% of the fans attending the game are Penn State fans, while 20% are Notre Dame fans. That is, \(P(P) = 0.8\) and \(P(N) = 0.2\). Then, by independence:
\(P(X = 0) = P(NNN) = 0.2 \times 0.2 \times 0.2 = 1 \times (0.8)^0\times (0.2)^3\)
And, by independence and mutual exclusivity of \(NNP\), \(NPN\), and \(PNN\):
\(P(X = 1) = P(NNP) + P(NPN) + P(PNN) = 3 \times 0.8\times 0.2\times 0.2 = 3\times (0.8)^1\times (0.2)^2\)
Likewise, by independence and mutual exclusivity of \(PPN\), \(PNP\), and \(NPP\):
\(P(X = 2) = P(PPN) + P(PNP) + P(NPP) = 3\times 0.8 \times 0.8 \times 0.2 = 3\times (0.8)^2\times (0.2)^1\)
Finally, by independence:
\(P(X = 3) = P(PPP) = 0.8\times 0.8\times 0.8 = 1\times (0.8)^3\times (0.2)^0\)
Do you see a pattern in our calculations? It seems that, in each case, we multiply the number of ways of obtaining \(x\) Penn State fans first by the probability of \(x\) Penn State fans \((0.8)^x\) and then by the probability of \(3x\) Nebraska fans \((0.2)^{3x}\).
This example lends itself to the creation of a general formula for the probability mass function of a binomial random variable \(X\).
 Binomial Random Variable \(X\)

The probability mass function of a binomial random variable \(X\) is:
\(f(x)=\dbinom{n}{x} p^x (1p)^{nx}\)
We denote the binomial distribution as \(b(n,p)\). That is, we say:
\(X\sim b(n, p)\)
where the tilde \((\sim)\) is read "as distributed as," and \(n\) and \(p\) are called parameters of the distribution.
Let's verify that the given p.m.f. is a valid one!
Now that we know the formula for the probability mass function of a binomial random variable, we better spend some time making sure we can recognize when we actually have one!