Question

2. . A random sample of 250 juniors majoring in psychology or communication at a large university is selected. These students are asked whether or not they are happy with their majors. The following table gives the results of the survey. Assume that none of these 250 students is majoring in both areas.

Table 1

	Happy (H)	Unhappy (U)
Psychology (P)	80	20
Communication (C)	115	35

(a). If one student is selected at random from this group, find the probability that this student is:

(i) happy with the choice of major.                                                                          

(ii) a communication major given that the student is happy with the choice of major.                                                                                                                                            

(iii) unhappy with the choice of major given that the student is a psychology major

(iv) a psychology major and is happy with that major.                                           

(v) a communication major or is unhappy with his or her major.                           

(b) Are the events “psychology major” and “happy with major” independent? Are they mutually exclusive? Explain why or why not.                     

Answer 1

Solution :

Probability of an event E is given as follows :

P(E) = number of favourable outcomes/Total number of outcomes

a(i) We have to find P(happy)

P(happy) = Number of students who are happy with the choice of major/Total number of students

P(happy) = (115 + 80)/250

P(happy) = 195/250

P(happy) = 0.78

Hence, the required probability is 0.78.

a(ii) We have to find P(Communication | happy)

P(communication and happy) = Number of students who have communication major and also they are happy with the choice of major/Total number of students

P(communication and happy) = 115/250

From previous part we have, P(happy) = 195/250

Hence, the required probability is 0.5897.

a(iii) We have to find P(unhappy | Psychology)

P(Unhappy and Psychology) = Number of students who are unhappy with the choice of major and they have Psychology major/Total number of students

P(Unhappy and Psychology) = 20/250

P(Psychology) = Number of students who have Psychology major/Total number of students

P(Psychology) = (80 + 20)/250

P(Psychology) = 100/250

Hence, the required probability is 0.20.

a(iv) We have to find P(Communication or Unhappy)

P(Communication or Unhappy) = P(Communication) + P(Unhappy) - P(Communication and Unhappy)

Now,

P(Communication) = (115 + 35)/250

P(Communication) = 150/250

P(Unhappy) = 55/250

P(Communication and Unhappy) = 35/250

Hence,

P(Communication or Unhappy) = (150/250) + (55/250) - (35/250)

P(Communication or Unhappy) = 170/250

P(Communication or Unhappy) = 0.68

Hence, the required probability is 0.68.

b) The events “psychology major” and “happy with major” are said to be independent iff,

P(Psychology and Happy) = P(Psychology).P(Happy)

We have,

P(Psychology and Happy) = 80/250

P(Psychology) = (80 + 20)/250 = 100/250

P(Happy) = 195/250

P(Psychology).P(Happy) = (100/250) × (195/250)

P(Psychology).P(Happy) = 39/125

Hence, the events “psychology major” and “happy with major” are not independent.

The events “psychology major” and “happy with major” are said to be mutually exclusive if

P(Psychology and Happy) = 0

We have, P(Psychology and Happy) = 80/250

Hence, the events “psychology major” and “happy with major” are not mutually exclusive.

Please rate the answer. Thank you.