Blue Eyes Assume that 35% of us have blue eyes (based on a study by Dr. P. Soria at Indiana University).

a. Let B denote the event of selecting someone who has blue eyes. What does the event \(\overline {\bf{B}} \) denote?

b. Find the value of P(\(\overline {\bf{B}} \)).

c. Find the probability of randomly selecting three different people and finding that all of them have blue eyes.

d. Find the probability that among 100 randomly selected people, at least 40 have blue eyes.

e. If 35% of us really do have blue eyes, is a result of 40 people with blue eyes among 100 randomly selected people a result that is significantly high?

Percentage of people with blue eyes is 35%

a. Let B be the event of selecting a person with blue eyes.

Then,\(\overline B \)is the complement of event B,where\(\overline B \)denotes the event of selecting some one who does not have blue eyes.

b. The relationship between probabilities of complementary events is:

\(P\left( {\overline B } \right) = 1 - P\left( B \right)\)

As the percentage of people with blue eyes is 35%, the probability of selecting a person with blue eye is 0.35. That is,\(P\left( B \right) = 0.35\).

Substitute the value,

\(\begin{aligned}{c}P\left( {\overline B } \right) = 1 - P\left( B \right)\\ = 1 - 0.35\\ = 0.65\end{aligned}\)

Therefore, \(P\left( {\overline B } \right) = 0.65\).

c. Let event E be the probability of selecting three different persons with blue eyes.

As each person is independent, the probability of event E is the product of probabilities for an individual person to have a blue color.

\(\begin{aligned}{c}P\left( E \right) = {\left( {P\left( B \right)} \right)^3}\\ = {\left( {0.35} \right)^3}\\ = 0.0429\end{aligned}\)

Therefore, the probability of selecting three different persons and finding that all of them have blue eyes is 0.0429.

d. Define X as the random variable for the number of peoplein 100 with blue eyes.

As each person is independent and has the same probability for having blue eyes, X is assumed to follow a binomial distribution with nequal to 100 and p as 0.35.

The criteria for normal approximation of binomial distribution is:

\(\begin{aligned}{c}np = 100\left( {0.35} \right)\\ = 35\\ > 10\end{aligned}\)

\(\begin{aligned}{c}n\left( {1 - p} \right) = 100\left( {1 - 0.35} \right)\\ = 65\\ > 10\end{aligned}\)

Thus, X can be approximated to normal.

Use Normal approximation for binomial distribution.

The parameters are,

\(\begin{aligned}{c}\mu = np\\ = 100 \times 0.35\\ = 35\\\end{aligned}\)

\(\begin{aligned}{c}\sigma = \sqrt {n \times p \times q} \\ = \sqrt {100 \times 0.35 \times 0.65} \\ = 4.770\end{aligned}\)

Thus, \(X \sim N\left( {35,{{4.769}^2}} \right)\) .

Required probability is\(P\left( {X \ge 40} \right) = P\left( {X > 39.5} \right)\)obtained using continuity correction.

The z-score associated to the value is:

\(\begin{aligned}{c}z = \frac{{39.5 - 35}}{{4.770}}\\ = 0.9434\end{aligned}\)

Thus, the required probability is:

\(\begin{aligned}{c}P\left( {Z > 0.9434} \right) = 1 - P\left( {Z < 0.9434} \right)\\ = 1 - 0.8273\\ = 0.1727\end{aligned}\)

Refer to standard normal table for the cumulative probability of the value 0.9434 obtained as 0.8273.

The probability that at least 40 people have blue eyes is 0.1727.

e) A result is said to be significantly high if the chances of getting the value or greater have a probability less than or equal to 0.05.

In this case,\(P\left( {X \ge 40} \right) = 0.1727\), which is greater than 0.05.

The result is significantly high because the values are not less than 0.05.