Question

identifying two independent events
brown law firm collected data on the transportation
choices of its employees for their morning commute. the
table shows the percentages of the type of transportation
of the male and female employees.
consider the following events:
a: the employee is male.
b: the employee is female.
c: the employee takes public transportation.
d: the employee takes his/her own transportation.
e: the employee takes some other method of
transportation.
which two events are independent?
a and c
a and d
b and d
b and e

Explanation:

Step1: Calculate probability of \(A\)

\(P(A)=\frac{36}{60} = 0.6\)

Step2: Calculate probability of \(C\)

\(P(C)=\frac{20}{60}=\frac{1}{3}\approx0.33\)

Step3: Calculate \(P(A\cap C)\)

\(P(A\cap C)=\frac{12}{60} = 0.2\)

Step4: Check \(P(A)\times P(C)\)

\(P(A)\times P(C)=0.6\times\frac{1}{3}=0.2\)
Since \(P(A\cap C)=P(A)\times P(C)\), \(A\) and \(C\) are independent.

Step5: Calculate probability of \(A\)

\(P(A) = 0.6\)

Step6: Calculate probability of \(D\)

\(P(D)=\frac{30}{60}=0.5\)

Step7: Calculate \(P(A\cap D)\)

\(P(A\cap D)=\frac{20}{60}=\frac{1}{3}\approx0.33\)

Step8: Check \(P(A)\times P(D)\)

\(P(A)\times P(D)=0.6\times0.5 = 0.3
eq\frac{1}{3}\)

Step9: Calculate probability of \(B\)

\(P(B)=\frac{24}{60}=0.4\)

Step10: Calculate probability of \(D\)

\(P(D) = 0.5\)

Step11: Calculate \(P(B\cap D)\)

\(P(B\cap D)=\frac{10}{60}=\frac{1}{6}\approx0.17\)

Step12: Check \(P(B)\times P(D)\)

\(P(B)\times P(D)=0.4\times0.5=0.2
eq\frac{1}{6}\)

Step13: Calculate probability of \(B\)

\(P(B)=0.4\)

Step14: Calculate probability of \(E\)

\(P(E)=\frac{10}{60}=\frac{1}{6}\approx0.17\)

Step15: Calculate \(P(B\cap E)\)

\(P(B\cap E)=\frac{6}{60}=0.1\)

Step16: Check \(P(B)\times P(E)\)

\(P(B)\times P(E)=0.4\times\frac{1}{6}\approx0.07
eq0.1\)

Answer:

A. A and C