91Ó°ÊÓ

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:

$$\begin{gathered} P(X=7)=\underset{x=1}{\overset{7}{∑}}{(1-0.40)}^{x-1}×0.40 \\ =0.0187 \end{gathered}$$

The probability that you must ask three or four people is

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