Question

Select three fans randomly at a football game in which Penn State is playing over Notre Dame. Identify whether the fan is Penn State fan (P) or a Notre Dame fan (D). Let X be the number of Penn State fans.

Answer 1

The fan is a Penn State fan $(P)$ or a Notre Dame fan $(N).$ This experiment yields the following sample space:

$$S=\{PPP, PPN,PNP, PNN, NPP, NPN, NNP, NNN\}$$

Let $X=$ the number of Penn State fans selected. The possible values of $X$ are, therefore, either $0,1,2,$ or $3.$

Since the game is a home game, let's 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:

$$P(X=0)=P(NNN)=(0.2)^3=0.008$$

By independence and mutual exclusivity of $NNP, NPN,$ and $PNN$

$$P(X=1)=P(NNP)+P(NPN)+P(PNN)$$

$$=3(0.8)(0.2)^2=0.096$$

By independence and mutual exclusivity of $PPN, PNP,$ and $NPP$

$$P(X=2)=P(PPN)+P(PNP)+P(NPP)$$

$$=3(0.8)^2(0.2)=0.384$$

By independence:

$$P(X=3)=P(PPP)=(0.8)^3=0.512$$

Check

$$0.008+0.096+0.384+0.512=1$$

The probability of the sample space is $1.$

Because the values that it takes on are random, the variable $X$ is a random variable (rv).