Suppose that the probability that an adult in America will watch the Super Bowl is 40%. Each person is considered independent. We are interested in the number of adults in America we must survey until we find one who will watch the Super Bowl.

a. In words, define the random variable X.

b. List the values that X may take on.

c. Give the distribution of X. X ~ _____(_____,_____)

d. How many adults in America do you expect to survey until you find one who will watch the Super Bowl?

e. Find the probability that you must ask seven people.

f. Find the probability that you must ask three or four people

In a Bernoulli trial, the likelihood of the number of successive failures before a success is obtained is represented by a geometric distribution, which is a sort of discrete probability distribution. A Bernoulli trial is a test that can only have one of two outcomes: success or failure.

Random variable in simple terms generally refers to variables whose values are unknown, therefore, in this case X is the number of adults in America that is surveyed until we find the one who will watch the Super Bowl.

Make the list of values that you want to use X may take on.

As we can see there is an upper bound for the situation at hand so,

$X=1,2,3,4,........$

The random variable X refers to the number of trials before the first success. Each trial is independent of others and has similar probability of success.

This implies that random variable X follows Geometric Distribution.

Thus, the distribution of X is$X~G(0.40)$

The expected value of geometric distribution is:

$E\left(X\right)=\frac{1}{p}$where$p=0.40$

Therefore,

$$\begin{gathered} E\left(X\right)=\frac{1}{p} \\ E\left(X\right)=\frac{1}{0.40} \\ E\left(X\right)=2.5 \end{gathered}$$

The probability that you must ask seven people will be:

$$\begin{gathered} P(X=7)=\sum _{x=1}^7{(1-0.40)}^{x-1}\times 0.40 \\ =0.0187 \end{gathered}$$

The probability that you must ask three or four people is

$$\begin{gathered} P(X=3orX=4)=P(X=3)+P(X=4) \\ {=\left[\right(1-0.40)}^{3-1}\times (0.40)]+[{(1-0.40)}^{4-1}\times (0.40)] \\ =0.144+0.0864 \\ =0.2304 \end{gathered}$$