Question

The accompanying table categorizing each student in a sample according to gender and eye color appeared in the article "Does Eye Color Depend on Gender? It Might Depend on Who or How You Ask"

Suppose that one of these 2026 students are randomly selected. Let F denote the event that the selected individual is a female, and A, B, C, and D represent the events that he or she has blue, brown, green, and hazel eyes, respectively.
Eye colour
Gender Blue Brown Green Hazel Total
Male 370 352 198 187 1107
Female 359 290 110 160 919
Total 729 642 308 347 2026
A. Calculate both P(F) and P(C).
B. Calculate P(F C). Are the events F and C independent? Why or why not?
C. If the selected individual has green eyes, what is the probability that he or she is a female?
D. If the selected individual is female, what is the probability that she has green eyes?

Answer 1

Kindly check explanation

Explanation:

Given the data:

Gender Blue _ Brown _ Green _ Hazel _ Total

Male __ 370 _ 352 ___ 198 ___ 187 ___ 1107

Female_ 359 _ 290 ___ 110 ___ 160 ___ 919

Total __ 729__ 642 ___ 308 __ 347___ 2026

P(F) - probability of female

P(A) - Probability of blue

P(B) - Probability of brown

P(C) - Probability of green

P(D) - Probability of hazel

A. Calculate both P(F) and P(C).

P(F) = total Females /total number of students

P(F) = 919 / 2026 = 0.454

P(C) = total number of green eye / total students

P(C) = 308 / 2026 = 0.152

B. Calculate P(FnC). Are the events F and C independent? Why or why not

P(FnC) = number of green-eyed female / total students

P(FnC) = 110 / 2026 = 0.054

For independent event :

P(F) * P(C) = P(FnC)

0.454 * 0.152 = 0.069

P(FnC) = 0.054

Hence, P(F) * P(C) ≠ P(FnC) ; hence event aren't independent

C. If the selected individual has green eyes, what is the probability that he or she is a female?

P(F|C) = P(FnC) / P(C)

P(F|C) = (110 / 2026) ÷ (308/2026)

P(F|C) = 110/2026 * 2026/308

P(F|C) = 110/308 = 0.357

D. If the selected individual is female, what is the probability that she has green eyes?

P(C|F) = P(FnC) / P(F)

P(C|F) = 110/2026 ÷ 919/2026

P(C|F) = 110/2026 * 2026/919

P(C|F) = 110/919

= 0.121