# 10.1 - The Probability Mass Function

10.1 - The Probability Mass Function

## Example 10-1 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 $$3-x$$ Nebraska fans $$(0.2)^{3-x}$$.

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:

$$f(x)=\dbinom{n}{x} p^x (1-p)^{n-x}$$

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!

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