Question

A study is conducted to survey (in thousands) of earned degrees in the United States in a recent year. The table is given below.

	AA degree	Bachelor’s	Master’s	Professional	Total
Female	834	616	194	30	1674
Male	726	529	17	44	1316
Total	1560	1145	211	74	2990

a) If one person is randomly selected, find the probability that this person is a female.      

  b)  If one person is randomly selected, find the probability that this person has a bachelor degree and is a male.   

  c)   If one person is randomly selected, find the probability that this person has an AA degree.  

d) If one person is randomly selected, find the probability that this person is a female, giventhat the person received an AA degree.

e) If one person is randomly selected, find the probability that this person has a master degree or is a female.

f) Are the events “female” and “AA degree” independent? Why or why not? Use the answers from a) and d) above to explain this.

g) If two people are randomly selected, find the probability that these two people are males.

h) If one person is randomly selected, are the events “master degree” and female mutually exclusive? Why or why not? Explain clearly.

Answer 1

solution:

Given data

Gender	AA Degree	Bachelor's	Master's	Professional	Total
Female	834	616	194	30	1674
Male	726	529	17	44	1316
Total	1560	1145	211	74	2990

Given Total No.of people = n(S) = 2990

a) Let F = event of selected person is Female

Total No.of Female = n(F) = 1674

P (selected person is Female ) = n(F) / n(S) = 1674 / 2990 = 0.559

   Probability that selected person is Female = 0.559

b) Let A = event of selected person is male and has Bachelor Degree

No.of persons who is male and have Bachelor's Degree = n(A) = 529

P( selected person is male and has Bachelor's Degree ) = n(A) / n(S) = 529 / 2990 = 0.177

c) Let B = event of selected person has AA Degree

Total No.of Persons having AA Degree = n(B) = 1560

P (selected person is Female ) = n(B) / n(S) = 1560 / 2990 = 0.522

   Probability that selected person has an AA Degree= 0.522

d) Probability of selected person is Female,Given that person received an AA Degree = P (F | B) = P(F B) / P(B)

Total No.of Females who received AA Degree = n(F ​​​​​​​ B) = 834

P (selected person is Female ,Given that person has AA Degree ) = P(F ​​​​​​​ B) / P(B)

=(834/2990) / (1560/2990)

=834 / 1560

= 0.535

Probability of selected person is Female,Given that person received an AA Degree = 0.535

e) Let C = event of selected person has Master's Degree

n(C) = 211 , n(F) = 1674 , No.of females who received Master's Degree = n(C  ​​​​​​​ F) = 194

P( selected person has Master's Degree or is a Female ) = P( C U F)

=P(C) + P(F) - P(C  ​​​​​​​ F)

=(211/ 2990) +(1674/2990) - (194/2990)

= 0.566

   Probability of selected person has Master's Degree (or) is a Female = 0.566

f) we need to check F and B events Independent or not.

If F and B are Independent then P(F| B) = P(F)

From a) we have P(F) = 0.559 , from d) we have P(F| B) = 0.535

     P(F| B) != P(F)

    F (Female ) and B (AA Degree) events are not Independent

g) Let M = event of selected 2 persons are males

Total No.of males = 1316

n(M) =  1316 C 2 = 865270

Here , n(S) = 2990C2 = 4468555

P( selected 2 persons are males) = n(M) / n(S) = 865270 / 4468555 = 0.194

h) we need to check events C (Master's Degree) and F (Female) are mutually exclusive (Or) not

If C and F are Mutually exclusive then P(C  ​​​​​​​ F) = 0 {i.e., No Female must not have Master's Degree]

But 194 Females has Master's Degree.  

So, From e) we have  P(C  ​​​​​​​ F) = 194 / 2990 = 0.065 ! =0

   Events C (Master's Degree) and F (Female) are not mutually exclusive