College Algebra, Chapter 10, 10.4, Section 10.4, Problem 24

In a certain country, $20 \%$ of the population have a college degree. A jury consisting of 12 people is selected at random from this country.

a.) What is the probability that exactly two of the jurors have a college degree?

b.) What is the probability that three or more of the jurors have a college degree?

Recall that the formula for the binomial probability is given by

$C(n,r) p^r q^{n-r}$

In this case, the probability of success is $p =0.20$ and the probability of failure is $q = 1-p = 0.80$.

a.) The probability that exactly two of the jurors where $r=2$ have a college degree is

$= C(12,2)(0.20)^2 (0.80)^{12-1}$

$= 0.2835$

b.) To solve this more easily, we will apply the complement to the probability that none, 1 or 2 out of the 12 people selected has a college degree. Thus, we get


$
\begin{equation}
\begin{aligned}

p(\text{3 or more jurors have college degree}) =& 1 - [p(0) + p(1) +
p(2)]
\\
\\
=& 1 - \left[C(12,0) (0.20)^0 (0.80)^{12-0} + C(12,1) (0.20)^1 (0.80)^{12-1} + C(12,2) (0.20)^2 (0.80)^{12-2}\right]
\\
\\
=& 1 - [0.0687 + 0.2062 + 0.2835]
\\
\\
=& 0.4416
\end{aligned}
\end{equation}
$