Question

One class has 30 students. Ten are women (F) and US citizens (U); 12 are male (M) and US citizens; 6 are women who are not US citizens (N); 2 are men who are not US citizens. A name is selected at random from the class list and it is female. Determine the following probabilities:

1. P (FU)
2. P (FN)
3. P (MU)
4. P (MN)
5. P (F)
6. P (M)
7. P (U)
8. P (N)
9. P (U/F)

Answer 1

One class has 30 students. Therefore, total number of students = 30

Ten are women (F) and US citizens (U). Therefore, number of women who are US citizens = 10

12 are male (M) and US citizens (U). Therefore, number of males who are US citizens = 12

6 are women who are not US citizens (N). Therefore, number of women who are not US citizens = 6

2 are men who are not US citizens. Therefore, number of males who are not US citizens = 2

Writing this data in the form of a table :

	U	N	Total
F	10	6	16
M	12	2	14
Total	22	8	30

a. Answer :

P(FU) = number of women who are US citizens / total number of students

           = 10 / 30

           = 0.33

Therefore, P(FU) = 0.33

b. Answer :

P(FN) = number of women who are not US citizens / total number of students

           = 6 / 30

           = 0.20

Therefore, P(FN) = 0.20

c. Answer :

P(MU) = number of males who are US citizens / total number of students

           = 12 / 30

           = 0.40

Therefore, P(MU) = 0.40

d. Answer :

P(MN) = number of males who are not US citizens / total number of students

            = 2 / 30

            = 0.07

Therefore, P(MN) = 0.07

e. Answer :

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

        = 16 / 30

        = 0.53

Therefore, P(F) = 0.53

f. Answer :

P(M) = number of males / total number of students

         = 14 / 30

         = 0.47

Therefore, P(M) = 0.47

g. Answer :

P(U) = number of US citizens / total number of students

        = 22 / 30

        = 0.73

Therefore, P(U) = 0.73

h. Answer :

P(N) = number of students who are not US citizens / total number of students

        = 8 / 30

        = 0.27

Therefore. P(N) = 0.27

i. Answer :

P(U | F) = P(FU) / P(F)   ............(conditional probability)

              = 0.33 / 0.53      .............(from part a. and part e.)

              = 0.62

Therefore, P(U | F) = 0.62