Question

Assume that in a statistics class 48% of the students are in third year. Further, 49% of the students in third year are Male, and 53% of the students in a year other than third are Female.

A. If a student is randomly selected, what is the probability she is Female and is in third year?

B. If a student is randomly selected, what is the probability he is Male but not in third year?

C. If a student is randomly selected, what is the probability that he is Male?

D. Are the events Male and not in third year mutually exclusive? Explain.

E. Are the events Female and third year independent? Explain.

Answer 1

Based on data given following table can be prepared

	Third Year	Not Third Year
Male	=0.59*48 = 23.52	= 52-27.56 = 24.44	= 23.52+24.44 = 47.96
Female	= 48-23.52 = 24.48	=0.53*52 = 27.56	= 24.48+27.56 = 52.04
	48	= 100-48 = 52	100

Table has been prepared in following manner:

Here we have assumed that there are 100 students in statistics class. Since 48% students in statistics class are in third year. So sum of rows in third year column will be 48. The remaining 100-40 = 52 students are not in third year. So sum of rows in not third year column will be 52. Now, it is given that 49% of the students in third year are Male. So no. of males in third year = 0.49*48 = 23.52. So no. of females in third year = 48-23.52 = 24.48. Now, it is given that 53% of the students in a year other than third are Female. So no. of females not in third year = 0.53*52 = 27.56. So no. of males not in third year = 52-27.56 = 24.44.

Now all questions can be answered from table directly.

Answer A)

Probability randomly selected student is Female and is in third year = No. of females in third year/Total no. of students

Probability randomly selected student is Female and is in third year = 24.48/100 (Refer table below)

Probability randomly selected student is Female and is in third year = 0.2448

Answer B)

Probability randomly selected student is male and not in third year = No. of males not in third year/Total no. of students

Probability randomly selected student is male and not in third year = 24.44/100 (Refer table below)

Probability randomly selected student is male and not in third year = 0.2444

Answer C)

Probability randomly selected student is male = Total no. of males in statistic class/Total no. of students

Probability randomly selected student is male = 47.96/100 (Refer table below)

Probability randomly selected student is male = 0.4796

Answer D)

Two events A and B are said to be mutually exclusive events when both cannot occur at the same time.

For mutually exclusive events P(A⋂B) = 0

The events Male and not in third year are not mutually exclusive because in this case, P(Male⋂Not in Third Year) is 0.2444 (Refer part C)

Answer E)

When two events, A and B, are independent, the probability of both occurring is:

P(A and B) = P(A) · P(B)

P(Female) = 52.04/100 = 0.5204

P(Third Year) = 48/100 = 0.48

P(Female and Third Year) = 0.2448

P(Female)*P(Third Year) = 0.5204*0.48 = 0.2498

Since P(Female and Third Year) ≠  P(Female)*P(Third Year) so two events are not independent

The events Female and third year are not independent